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 $L(s,\pi )$ be its L-function. We introduce a universal lower bound for the integral ${\int }_{-\infty }^{+\infty }|\frac{L(\frac{1}{2}+it,\pi )}{\frac{1}{2}+it-s}{|}^{2}dt$ where s is equal to 0 or is a zero of $L\left(s\right)$ on the critical line. In the main text, the proof is given for $m\le 2$ 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 $L(s,\pi )$. In the Appendix, written with Peter Sarnak, the proof is made unconditional for general m.