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

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,π,Sym^2\otimesχ)$ of an irreducible cuspidal automorphic representation $π$ of $\GL_r(\A)$ twisted by a Hecke character $χ$. In particular, we will show that the $L$-function $L^S(s,π,Sym^2\otimesχ)$ is holomorphic except at $s=0$ and $s=1$, and moreover the possible poles could occur only when $χ^rω^2=1$, 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 $χ=1$.

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