In this paper, we consider the (partial) symmetric square -function of an irreducible cuspidal automorphic representation of twisted by a Hecke character . In particular, we will show that the -function is holomorphic for the region with the exception that, if , a pole might occur at , where is the central character of . Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case .
Shuichiro Takeda. "The twisted symmetric square -function of ." Duke Math. J. 163 (1) 175 - 266, 15 January 2014. https://doi.org/10.1215/00127094-2405497