ar X iv : m at h / 03 06 05 2 v 2 [ m at h . N T ] 2 A pr 2 00 4 Effective multiplicity one on GL ( n )

We give a narrow zero-free region for standard L-functions on GL(n) and Rankin-Selberg L-functions on GL(m) × GL(n) through the use of positive Dirichlet series. Such zero-free regions are equivalent to lower bounds on the edge of the critical strip, and in the case of L(s, π × ˜ π), on the residue at s = 1. Using the latter we show that a cuspidal automorphic representation on GL(n) is determined by a finite number of its L-series coefficients, and that this number grows at most polynomially in the analytic conductor.