A weighted one-level density of families of $L$-functions

This paper is devoted to a weighted version of the one-level density of the non-trivial zeros of L-functions, tilted by a power of the L-function evaluated at the central point. Assuming the Riemann Hypothesis and the ratio conjecture, for some specific families of L-functions we prove that the same structure suggested by the density conjecture holds also in this weighted investigation, if the exponent of the weight is small enough. Moreover we speculate about the general case, conjecturing explicit formulae for the weighted kernels. 1. A weighted version of the one-level density Let us assume the Riemann Hypothesis for all the L-functions that arise. The classical one-level density considers a smooth localization at the central point of the counting function of the non-trivial zeros of an L-function, averaged over a “natural” family of Lfunctions in the Selberg class. More specifically, given an even and real-valued function f in the Schwartz space and an L-function L(s) in a family F , we consider the quantity (1.1) ∑ γL f(c(L)γL) where γL denotes the imaginary part of a generic non-trivial zero of L and c(L) the logconductor of L(s) at the central point. We recall that 1/c(L) is the mean spacing of the non-trivial zeros of L(s) around s = 1 2 . The one-level density for the family F is the average of the above quantity over the family, i.e. (1.2) lim X→∞ 1 ∑ L∈FX 1 ∑ L∈FX ∑