# RMA

## Seminar

### Bounds at s=1 for an axiomatic class of L-functions

Seminar Room 1, Newton Institute

#### Abstract

When the Ramanujan hypothesis about the Dirichlet coefficients of a generic L-function is assumed, it is quite easy to prove upper-bounds of type L(1)<< R^c, for every c>0, where R is a parameter related to the functional equation of L. We show how to prove the same bound when the Ramanujan hypothesis is replaced by a much weaker assumption and L has Euler product of polynomial type. As a consequence, we obtain an upper bound of this type for every cuspidal automorphic GL(n) L-function, unconditionally. We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the symmetric cube of a Maass form.