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

Abstract

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,\pi ,{Sym}^2\otimes \chi )$ of an irreducible cuspidal automorphic representation $\pi$ of ${GL}_r\left(\mathbb{A}\right)$ twisted by a Hecke character $\chi$. In particular, we will show that the $L$-function $L^S(s,\pi ,{Sym}^2\otimes \chi )$ is holomorphic for the region $Re\left(s\right)>1-1/r$ with the exception that, if ${\chi }^r{\omega }^2=1$, a pole might occur at $s=1$, where $\omega$ is the central character of $\pi$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $\chi =1$.