A specialisation of the Bump-Friedberg $L$-function

Abstract

We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+α,2s),s\in\C\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product $L(s+α,π)L(2s,Λ^2,π)$ which we denote by $L^{lin}(s,π,α)$ for this abstract, when $π$ is a cuspidal automorphic representation of $GL(k,A)$ for $A$ the adeles of a number field. When $k$ is even, we show that for a cuspidal automorphic representation $π$, the partial $L$-function $L^{lin,S}(s,π,α)$ has a pole at 1/2, if and only if $π$ 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 $L(α+1/2,π)\neq 0$ and $L(1,Λ^2,π)=\infty$. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $Re(s)\geq 1/2$ when $Re(α)$ is $\geq 0$.

The paper is under the form that will appear in Canad. Math. Bull