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

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.