The twisted symmetric square L-function of GL(r)

Abstract

Abstract. In this paper, we consider the (partial) symmetric square L-function L S (s, π, Sym 2 ⊗ χ) of an irreducible cuspidal automorphic representation π of GLr(A) twisted by a Hecke character χ. In particular, we will show that the L-function L S (s, π, Sym 2 ⊗ χ) is holomorphic for the region Re(s) > 1 − 1 r with the exception that, if χ r ω 2 = 1, a pole might occur at s = 1, where ω is the central character of π. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case χ = 1.

Doubling constructions and tensor product L-functions: the linear case

“…where Λ ν is a scalar multiple of Λ 0 ν for all ν ∉ S. Proof. Similar to [Tak14,Proposition 3.14], which is an adaptation of the decomposition result when uniqueness holds everywhere (see [Sha74,§ 4], [Bum97, Theorem 3.5.2]).…”

“…This is the weaker form of an Eulerian integral we can obtain, called an "almost Euler product" by Takeda [Tak14].…”

The role of the Ramanujan conjecture in analytic number theory

“…For their part, the symmetric square L-functions proved more resistant to the explication of their analytic properties. From the work of many authors, most notably Bump and Ginzberg [21], Shahidi [132], Kim [72], and Takeda [141], one knows that L(s, π, sym 2 ) is nice, except possibly for some exceptional poles within the critical strip. In contrast to the Rankin-Selberg Lfunctions, the absolute convergence of L(s, π, sym 2 ) in s > 1 is only known for n 4.…”

On a certain metaplectic Eisenstein series and the twisted symmetric square L-function

“…, q, where each permutation corresponds to a permutation of GL 2 -blocks in the Levi GL 2 × · · ·×GL 2 × GL 1 . (Note that the last GL 1 is always fixed by w.) But by exactly the same reasoning as the case r = 2q, one can see that the Eisenstein series has a residue at ν = ρ P r−1,1 2,...,2,1 /2 only for w = (12 · · · q) by using [T1,Proposition 2.42].…”