A Specialisation of the Bump–Friedberg L-function

Abstract

AbstractWe study the restriction of Bump–Friedberg integrals to affine lines {(s + α, 2s), s ∊ ℂ}. It has 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, Λ2, π), which we denote by Llin(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 the partial L-function Llin,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) ≠ 0 and L(1, Λ2 , π) = ∞. When k is odd, the partial L-function is holmorphic in a neighbourhood of Re(s) ≥ 1/2 when Re(α) is ≥ 0.