  We establish zero-free regions tapering as an inverse power of the analytic
conductor for Rankin-Selberg L-functions on GL(n) x 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,f x f~), 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.
