A universal lower bound for certain quadratic integrals of automorphic L–functions

Abstract

Let π be a cuspidal unitary representation od GL(m,A) where A denotes the ring of adèles of Q. Let L(s,π) be its L-function. We introduce a universal lower bound for the integral ∫−∞+∞|L(12+it,π)12+it−s|2dt where s is equal to 0 or is a zero of L(s) on the critical line. In the main text, the proof is given for m≤2 and under a few assumptions on π. It relies on the Mellin transform; the proof involves an extension of a deep result of Friedlander-Iwaniec. An application is given to the abscissa of convergence of the Dirichlet series L(s,π). In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.