While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form
$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D70C}}R(\unicode[STIX]{x1D70C})x^{\unicode[STIX]{x1D70C}},\end{eqnarray}$$
where the sum is over the non-trivial zeros
$\unicode[STIX]{x1D70C}$
of
$\unicode[STIX]{x1D701}(s)$
,
$R(x)\in \overline{\mathbb{Q}}(x)$
is a rational function over algebraic numbers and
$x>0$
is a real algebraic number. In particular, we show that the function
$$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$
has infinitely many zeros in
$(1,\infty )$
, at most one of which is algebraic. The transcendence tools required for studying
$f(x)$
in the range
$x<1$
seem to be different from those in the range
$x>1$
. For
$x<1$
, we have the following non-vanishing theorem: If for an integer
$d\geqslant 1$
,
$f(\unicode[STIX]{x1D70B}\sqrt{d}x)$
has a rational zero in
$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$
, then
$$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$
where
$\unicode[STIX]{x1D712}_{-d}$
is the quadratic character associated with the imaginary quadratic field
$K:=\mathbb{Q}(\sqrt{-d})$
. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.