We study the restriction of Bump–Friedberg integrals to affine lines
$\left\{ \left( s+\alpha ,2s \right),s\in \mathbb{C} \right\}$
. It has simple theory, very close to that of the Asai L-function. It is an integral representation of the product
$L\left( s+\alpha ,\pi \right)L\left( 2s,{{\Lambda }^{2}},\pi \right)$
, which we denote by
${{L}^{\operatorname{lin}}}\left( s,\pi ,\alpha \right)$
for this abstract, when
$\pi$
is a cuspidal automorphic representation of
$GL\left( k,\mathbb{A} \right)$
for
$\mathbb{A}$
the adeles of a number field. When
$k$
is even, we show that the partial
$L$
-function
${{L}^{\text{lin,S}}}\left( s,\text{ }\!\!\pi\!\!\text{ ,}\alpha \right)$
has a pole at
$1/2$
if and only if
$\pi$
admits a (twisted) global period. This gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that π has a twisted global period if and only if
$L\left( \alpha +1/2,\text{ }\!\!\pi\!\!\text{ } \right)\ne 0$
and
$L\left( 1,{{\Lambda }^{2}},\pi \right)=\infty $
. When
$k$
is odd, the partial
$L$
-function is holmorphic in a neighbourhood of
$\operatorname{Re}\left( s \right)\ge 1/2$
when
$\operatorname{Re}\left( \alpha \right)\,\,\text{is}\,\,\ge \text{0}\,$
.