2.1 Notation

We mention the following asymptotic notation.

1. (1) $f(s)={\mathcal O}(g(s)),~s \in S$ , or equivalently, $f(s) \ll g(s),~s \in S$ , means there exists a constant c such that $|f(s)| \leq c|g(s)|$ for all $s\in S$ .

2. (2) $f(s) \sim g(s),~ s\rightarrow s_0$ , means $\lim _{s\rightarrow s_0}{f(s)}/t{g(s)} = 1$ .

2.2 Preliminaries

Let $\mathbb {H}$ denote the upper half plane. For $z \neq 0$ and $s \in \mathbb {C}$ , we set $z^s = \exp (s \log z)$ with $\log z = \log |z| + i \arg z$ , where $-\pi < \arg z \leq \pi $ . Let $k \geq 12$ be a positive even integer and $z\in \mathbb {H}$ .

Let $S_k$ denote the space of cusp forms of weight k with respect to $\mathrm {SL}_2(\mathbb {Z})$ . For $g \in S_k$ , the associated L-series,

$$ \begin{align*}L(g,s):=\sum_{n \geq 1} \frac{a_g(n)}{n^s},\end{align*} $$

is holomorphic on the half plane $\mathrm {Re}(s)>{(k+1)}/{2}$ . We define the completed Hecke L-series associated to g as

$$ \begin{align*} L^*(g,s) := (2\pi)^{-s}\Gamma(s)L(g,s), \end{align*} $$

which is also equal to the Mellin transform of g, given by

$$ \begin{align*} \mathcal{M}(g)(s):=\int_{0}^{\infty}g(it)t^{s-1}\,dt. \end{align*} $$

Since $\mathcal {M}(g)(s)$ is known to be entire, $L^*(g,s)$ (and also $L(g,s)$ ) can be uniquely extended as an entire function. Moreover, $L^*(g,s)$ satisfies the functional equation

(2.1) $$ \begin{align} L^*(g,k-s) = (-1)^{{k}/{2}}L^*(g,s). \end{align} $$

For a normalised Hecke eigenform f in $S_k$ , we define the symmetric square L-function of f by the formula

$$ \begin{align*} L(\text{Sym}^2(f),s):=\prod_{p} \frac{1}{(1-\alpha_p^2p^{-s})(1-\alpha_p\beta_pp^{-s})(1-\beta_p^2p^{-s})}, \end{align*} $$

for $\mathrm {Re}(s)>k$ , where $\alpha _p$ and $\beta _p$ are the roots of the polynomial $X^2-a_f(p)X+p^{k-1}$ . The function $L(\text {Sym}^2(f),s)$ extends to an entire function [Reference Shimura15] which is invariant under the map $s\mapsto 2k-1-s$ .

For $x,y\in \mathbb {C}$ and $\mathrm {Re}(x),\mathrm {Re}(y)>0,$ we have the beta function

$$ \begin{align*} B(x,y) := \int_{0}^{1} t^{x-1}(1-t)^{y-1} \,dt \end{align*} $$

and we recall that it satisfies

(2.2) $$ \begin{align} B(x,y) &= \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= 2\int_{0}^{\pi/2} (\sin \theta)^{2x-1}(\cos \theta)^{2y-1} \,d\theta. \end{align} $$

2.3 Kohnen’s cusp form

Let $s\in {\mathbb C}$ with $1<\mathrm {Re}(s)<k-1$ . The kernel function $R_{k,s}(z)$ , introduced by Kohnen [Reference Khuri-Makdisi, Kohnen and Raji6], is defined by

(2.3) $$ \begin{align} R_{k,s}(z) := \gamma_{k}(s) {\sum}^{\prime} z^{-s}\bigg\vert_{k} \begin{pmatrix} a & b \\ c & d \end{pmatrix}. \end{align} $$

Here $\vert _{k}$ denotes the standard weight k (integer) action of $\mathrm {GL}_2^+({\mathbb Q})$ on the functions $g:\mathbb {H} \rightarrow \mathbb {C}$ , defined by

$$ \begin{align*} (g\vert_{k}\gamma)(\tau) := (ad-bc)^{k/2} (c\tau +d)^{-k} g\bigg(\dfrac{a\tau +b}{c\tau + d}\bigg), \quad \tau \in \mathbb{H}, \gamma = \begin{pmatrix} a&b\\c&d \end{pmatrix} \in GL_2^+(\mathbb{Q}), \end{align*} $$

and the sum $\sum ^\prime $ is taken over all matrices $(\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}) \in \mathrm {SL}_2(\mathbb {Z})$ . Moreover,

$$ \begin{align*}\gamma_k(s) := \tfrac{1}{2}e^{{\pi i s}/{2}}\Gamma(s)\Gamma(k-s). \end{align*} $$

The function $R_{k,s}(z)$ is a cusp form of weight k for the full modular group $\mathrm {SL}_2(\mathbb {Z})$ . If $\langle \,,\rangle $ denotes the usual Petersson inner product on $S_k$ , then the essential property of $R_{k,s}$ is that for any $g \in S_k$ , by [Reference Khuri-Makdisi, Kohnen and Raji6, Lemma 1],

(2.4) $$ \begin{align} \langle g,R_{k,\bar{s}}\rangle = c_kL^*(g,s), \quad \mbox{where } c_k := \frac{(-1)^{{k}/{2}}\pi(k-2)!}{2^{k-2}}. \end{align} $$

7.1 Estimating the main term

We have $0 < \epsilon _1 < \epsilon _1 + \delta ^{\prime } \leq \epsilon _2 \leq \tfrac 12$ , $|\beta _j| \leq T$ and $\sum _{j=1}^{2} \epsilon _j \geq \tfrac 12+\delta $ . (Note that this forces $\epsilon _1\geq \delta $ .) From (4.1),

$$ \begin{align*} \bigg|\frac{\Gamma(s_2)}{\Gamma(k-s_1)}\bigg| \geq \frac{1}{\sqrt{\cosh \pi\beta_2}}\frac{\Gamma(\frac{k}{2}+\epsilon_2)}{\Gamma(\frac{k}{2}-\epsilon_1)} \geq e^{-{\pi|\beta_2|}/{2}}\bigg(\frac{k}{2}+\epsilon_2-1\bigg)^{\epsilon_1+\epsilon_2} \geq \frac{k^{\epsilon_1+\epsilon_2}}{3}e^{-{\pi T}/{2}}. \end{align*} $$

Similarly, one can see that

$$ \begin{align*}\bigg|\frac{\Gamma(s_2)}{\Gamma(s_1)} \bigg| \leq e^{{\pi T}/{2}}k^{\epsilon_2-\epsilon_1}. \end{align*} $$

Next, we find a lower bound for $\zeta (s_1+s_2-k+1)$ . By Lemma (4.6),

$$ \begin{align*} |\zeta(s_1+s_2-k+1)| \geq \frac{\zeta(2(1+\epsilon_2+\epsilon_1))}{\zeta(1+\epsilon_2+\epsilon_1)}> \frac{\zeta(4)}{\zeta(1.5)} = c_1. \end{align*} $$

Now, the function $(x-1)\zeta (x)$ is bounded in the interval $1 \leq x \leq 2$ . So, set

$$ \begin{align*}c_2:=\displaystyle \max_{1\leq x \leq 2} (x-1)\zeta(x). \end{align*} $$

Then $|\zeta (s_2-s_1+1)| \leq \zeta (\epsilon _2-\epsilon _1+1) \leq \zeta (\delta ^{\prime }+1) \leq {c_2}/{\delta ^{\prime }}$ , since $\epsilon _2-\epsilon _1 \geq \delta ^{\prime }$ . One therefore sees that

(7.3) $$ \begin{align} &|L^*(T_{\mathrm{main},s_1},s_2)| \nonumber\\ &\quad \geq 2\bigg\{(2\pi)^{-(\epsilon_1+\epsilon_2)} \bigg|\frac{\Gamma(s_2)}{\Gamma(k-s_1)}\zeta(s_1+s_2-k+1)\bigg| -(2\pi)^{\epsilon_1-\epsilon_2}\bigg|\frac{\Gamma(s_2)}{\Gamma(s_1)}\zeta(\epsilon_2-\epsilon_1+1)\bigg|\bigg\}\nonumber\\ &\quad \geq \frac{c_1}{3\pi}k^{\epsilon_1+\epsilon_2} e^{-{\pi T}/{2}} - \frac{2c_2}{\delta^{\prime}}e^{{\pi T}/{2}} k^{\epsilon_2-\epsilon_1}. \end{align} $$

7.2 Estimating the error term

Next, we consider the term $T_{\mathrm {error},s_1}(it)$ . We claim that the component $T_{\geq 2,\geq 2,s_1}(it)$ is modular under the action of $S:=(\begin {smallmatrix} 0&-1\\ 1&0 \end {smallmatrix})$ , that is, for all $t>0$ ,

$$ \begin{align*}T_{\geq2,\geq2,s_1}(it) = (it)^{-k} T_{\geq2,\geq2,s_1}(i/t).\end{align*} $$

This follows since

(7.4) $$ \begin{align} T_{\geq2,\geq2,s_1}(it) &= e^{i{\pi}s_1/2} \displaystyle\sum_{a^2+b^2 \geq 2} \sum_{\substack{c^2+d^2 \geq 2\\ \det \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big) = 1}} (cit+d)^{-k}\bigg(\frac{ait+b}{cit+d}\bigg)^{-s_1}\nonumber\\ &=e^{i{\pi}s_1/{2}} \displaystyle\sum_{a^2+b^2 \geq 2} \sum_{\substack{c^2+d^2 \geq 2\\ \det \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \big)= 1}} (it)^{-k}(-d(i/t)+c)^{-k}\bigg(\frac{-b(i/t)+a}{-d(i/t)+c}\bigg)^{-s}\nonumber\\ &=(it)^{-k}T_{\geq 2,\geq 2,s_1}(i/t). \end{align} $$

Thus,

$$ \begin{align*}L^*(T_{\geq 2,\geq 2,s_1},s_2) = \int_{1}^{\infty} T_{\geq 2,\geq 2,s_1}(it)(t^{s_2}+(-1)^{k/2}t^{k-s_2})\,\frac{dt}{t} \end{align*} $$

which implies that

(7.5) $$ \begin{align} |L^*(T_{\geq 2,\geq 2,s_1},s_2)| & \leq 250e^{{\pi T}/{2}}\bigg(\int_{1}^{\infty} \frac{1}{t^{k-2}}(t^{\sigma_2}+t^{k-\sigma_2})\,\frac{dt}{t}\bigg)\nonumber\\ & \leq 250e^{{\pi T}/{2}}\bigg( \frac{1}{k-\sigma_2-2} + \frac{1}{\sigma_2-2}\bigg) \leq \frac{2000e^{{\pi T}/{2}}}{k} \end{align} $$

for all $k \geq 12$ . However, the other terms $T_{\geq 2,1,s}^{\mathrm {error}}(it)$ and $T_{1,\geq 2,s}^{\mathrm {error}}(it)$ in $T_{\mathrm {error},s}(it)$ are not invariant under S, so we estimate them individually. Note that

$$ \begin{align*} L^*(T_{\mathrm{error},s_1},s_2)=L^*(T_{\geq 2,\geq 2,s_1},s_2)+\bigg(\int_{0}^{1}+\int_{1}^{\infty}\bigg)(T_{\geq 2,1,s_1}^{\mathrm{error}}(it) + T_{1,\geq 2,s_1}^{\mathrm{error}}(it))t^{s_2-1}\,dt. \end{align*} $$

Here, using the bounds in (5.3) and (5.5), we get

(7.6) $$ \begin{align} \bigg|\int_{1}^{\infty} (T_{\geq 2,1,s_1}^{\mathrm{error}}(it) + T_{1,\geq 2,s_1}^{\mathrm{error}}(it))t^{s_2-1} \,dt \bigg| &\leq 10e^{{\pi T}/{2}}\bigg(\int_{1}^{\infty} t^{\sigma_2-1-k} \,dt\bigg)\leq\frac{40e^{{\pi T}/{2}}}{k}, \end{align} $$

again for all $k \geq 12$ . However, these estimates are not useful when $0<t<1$ . Therefore, we proceed as follows. From (4.1) and (5.4), we see that

$$ \begin{align*}|T_{1,\geq 2,s_1}^{\mathrm{error}}(it)|\leq \frac{2}{t^k} (\cosh{\pi \beta_1})^{{1}/{2}}\frac{(2\pi)^{k-\sigma_1}}{\Gamma(k-\sigma_1)}\sum_{n\geq 1} n^{k-\sigma_1-1} e^{{-2\pi n}/{t}}+\frac{2}{t^{\sigma_1}}.\end{align*} $$

From Lemma 4.4, it follows that there exist absolute constants $M_0$ and $K_0$ such that

$$ \begin{align*} &\sum_{n\geq 1} n^{k-\sigma_1-1} e^{-2\pi nu} \leq M_0 \bigg(\frac{k-\sigma_1}{e}\bigg)^{k-\sigma_1}\frac{1}{(2\pi u)^{k-\sigma_1}}, \end{align*} $$

for $k \geq K_0$ uniformly for $u \geq 1$ . (Here we have replaced ${1}/{t}$ by u.) Thus,

$$ \begin{align*} &\bigg|\int_{0}^{1} T_{1,\geq 2,s_1}^{\mathrm{error}}(it) t^{s_2-1}\,dt\bigg|\nonumber \\&\quad\leq 2 (\cosh{\pi \beta_1})^{{1}/{2}}\frac{(2\pi)^{k-\sigma_1}}{\Gamma(k-\sigma_1)}\int_{1}^{\infty} \sum_{n\geq 1} n^{k-\sigma_1-1} e^{-2\pi nu}u^{k-\sigma_2}\,\frac{du}{u}+\frac{2}{\sigma_2-\sigma_1}\nonumber \\ &\quad\ll e^{{\pi T}/{2}}\frac{1}{\Gamma(k-\sigma_1)}\bigg(\frac{k-\sigma_1}{e}\bigg)^{k-\sigma_1}\frac{1}{\sigma_2-\sigma_1}+\frac{2}{\sigma_2-\sigma_1}\\&\quad\ll e^{{\pi T}/{2}}\sqrt{(k-\sigma_1)}~~\frac{1}{\delta^{\prime}} \leq \frac{e^{{\pi T}/{2}}}{\delta^{\prime}}\sqrt{k} \end{align*} $$

where we have used $\sigma _2-\sigma _1 \geq \delta ^{\prime }$ and the fact that

$$ \begin{align*}\frac{1}{\Gamma(k-\sigma_1)}\bigg(\frac{k-\sigma_1}{e}\bigg)^{k-\sigma_1} \ll \sqrt{k-\sigma_1}. \end{align*} $$

The last inequality follows from Stirling’s estimates in Lemma 4.3(b). Thus,

(7.7) $$ \begin{align} \int_{0}^{1} T_{1,\geq 2,s_1}^{\mathrm{error}}(it) t^{s_2-1}\,dt ={\mathcal O}_{T,\delta^{\prime}} (k^{1/2}). \end{align} $$

Similarly, it follows that

(7.8) $$ \begin{align} \int_{0}^{1} T_{\geq 2,1,s_1}^{\mathrm{error}}(it) t^{s_2-1}dt \ll e^{{\pi T}/{2}}\sqrt{\sigma_1}~~\frac{1}{\sigma_2+\sigma_1-k} = {\mathcal O}_{T}(k^{1/2}). \end{align} $$

The implicit constant in (7.8) has no dependency on $\delta ^{\prime }$ since we assume $\sigma _1+\sigma _2-k> \tfrac 12$ . From (5.7), (7.4), (7.6), (7.7) and (7.8), it follows that $ L^*(T_{\mathrm {error},s_1},s_2) = {\mathcal O}_{T,\delta ^{\prime }}(\sqrt {k}). $ Along with the above and (7.2) and (7.3), we have

$$ \begin{align*} |L^*(f_{k,s_1},s_2)| &\geq \bigg| \frac{c_1}{3\pi} e^{-{\pi T}/{2}} k^{\epsilon_1+\epsilon_2} - \frac{2c_2}{\delta^{\prime}}e^{{\pi T}/{2}} k^{\epsilon_2-\epsilon_1} - C(T,\delta^{\prime})k^{1/2}\bigg|. \end{align*} $$

Thus, it follows that

(7.9) $$ \begin{align} L^*(f_{k,s_1},s_2) \gg_{T, \delta^{\prime}} k^{\epsilon_2+\epsilon_1}, \end{align} $$

for sufficiently large $k \gg _{T, \delta , \delta ^{\prime }} 1$ since $\epsilon _2+\epsilon _1 \geq \tfrac 12+\delta $ .

Given an arbitrary pair of points $s_1,s_2$ satisfying $\epsilon _1+\epsilon _2 \geq \tfrac 12+\delta $ on the strict right-hand side of the critical line, that is, $\{{k}/{2} < \mathrm {Re}(s_j)\leq {(k+1)}/{2}\}$ , one can always choose the point with smaller real part as the kernel parameter and the other one to be the Mellin transform parameter (unless both points lie on the same vertical line, in which case, clearly, our result does not hold). And, even in situations where either (or both) $s_1,~s_2$ assume values strictly to the left of the critical line, by virtue of (7.1), one can perform the appropriate reflection $s_j \mapsto k-s_j$ , thus effectively reducing it to a question about a pair of points on the right half of the critical strip. This proves the theorem.

From (6.1),

(7.10) $$ \begin{align} L^*(f_{k,s_1},s_2) &= \int_{1}^{\infty} f_{k,s_1}(it)(t^{s_2}+(-1)^{k/2}t^{k-s_2})\,\frac{dt}{t} \nonumber\\&= \frac{(-1)^{{k}/{2}}\pi \Gamma(k-1)}{2^{k-3}\Gamma(s_1)\Gamma(k-s_1)} \displaystyle\sum_{f \in \mathcal{B}_k} \dfrac{L^*(f,s_1)}{\langle f,f\rangle} L^*(f,s_2). \end{align} $$

From (7.9) and (7.10) we immediately observe that, given a pair of points $s_1,s_2$ in the critical strip such that $\epsilon _j$ satisfies the conditions in (1.2), there is at least one eigenform $f \in \mathcal {B}_k$ whose L-function is simultaneously nonvanishing at both $s_1$ and $s_2$ for $k \gg _{T, \delta , \delta ^{\prime }} 1$ . However, one may observe that

$$ \begin{align*}\sum_{f \in \mathcal{B}_k} \dfrac{L^*(f,s_1)}{\langle f,f\rangle} L^*(f,s_2) \end{align*} $$

is the first Fourier coefficient of $E_{s_1,k-s_1}^*(z,s_2)$ (as a function of $z \in \mathbb {H}$ ), and this sum is already known to be nonzero for sufficiently large k (see [Reference Choie, Kohnen and Zhang2, formula (3.1)]). This already guarantees the existence of such a Hecke eigenform as above. However, we improve, albeit partially, the result in [Reference Choie, Kohnen and Zhang2] by explicitly determining a lower bound for the number $N_k(s_1,s_2)$ in Corollary 3.4.