Effective multiplicity one on GL n and narrow zero-free regions for Rankin-Selberg L -functions

Abstract

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L -functions on GL n ×GL n' . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L ( s , π X), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GL n is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.