Lower bounds for L -functions at the edge of the critical strip

Abstract

We prove a coarse lower bound for L -functions of Langlands-Shahidi type of generic cuspidal automorphic representations on the line Re ( s ) = 1. We follow the path suggested by Sarnak using Eisenstein series and the Maass-Selberg relations. The bounds are weaker than what the method of de la Vallée Poussin gives for the standard L -functions of GL n , but are applicable to more general automorphic L -functions. Our Theorem answers in a strong form a conjecture posed by Gelbart and Shahidi [ J. Amer. Math. Soc. 14 (2001)], and sharpens and considerably simplifies the proof of the main result of that paper.