Conditional lower bounds on the distribution of central values in families of $L$-functions

Abstract

We establish a general principle that any lower bound on the non-vanishing of central L-values obtained through studying the one-level density of low-lying zeros can be refined to show that most such L-values have the typical size conjectured by Keating and Snaith. We illustrate this technique in the case of quadratic twists of a given elliptic curve, and similar results should hold for the many examples studied by Iwaniec, Luo, and Sarnak in their pioneering work ( 2000) on 1-level densities.