Siegel Modular Forms, L-Functions, and Satake Parameters Original Research Article

In this paper we take a first step towards a multiplicity-one result for Siegel modular forms on Spn(image). We study two L-functions associated to Siegel modular forms, the spinor zeta function in genus 2 and the standard zeta function for arbitrary genus. Both the spinor and standard zeta function are defined as products over all primes and we show that the factors for almost all primes determine the L-function. The study of these zeta functions naturally leads to the study of an invariant related to Siegel modular forms, Satake p-parameters. Our result equivalently states that for a simultaneous Hecke eigenform, the Satake parameters for almost all primes determine the Satake parameters for all primes up to an occasional variation in sign.