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

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In this paper, we consider the (partial) symmetric square L-function Lˢ(s,π,Sym²⊗χ) of an irreducible cuspidal automorphic representation π of ᵣ() twisted by a Hecke character χ. In particular, we will show that the L-function Lˢ(s,π,Sym²⊗χ) is holomorphic except at s=0 and s=1, and moreover the possible poles could occur only when χʳω²=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.