The Spin $L$-function on $\mathrm{GSp}_6$ for Siegel modular forms

TL;DR

This paper establishes a new integral representation for the Spin L-function on GSp_6, linking it to Siegel modular forms and deriving key properties like functional equations and pole finiteness.

Contribution

It introduces a Rankin-Selberg integral for the degree eight Spin L-function on GSp_6, applicable to automorphic representations from Siegel modular forms, and proves functional equations and pole finiteness.

Findings

Derived the functional equation for the Spin L-function.

Proved finiteness of poles for the completed L-function.

Established integral representation for automorphic forms on GSp_6.

Abstract

We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\pi$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $\Lambda(\pi,Spin,s)$ of $\pi$.