Low lying zeros of families of L-functions

Low-lying Zeros of Dihedral L-functions

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the L-functions L(s, ψ), where ψ is a character of the ideal class group of the imaginary quadratic field Q( √ −D) (D squarefree, D > 3, D ≡ 3 (mod 4)). We prove that, in the vicinity of the central point s = 1/2, the average distribution of these zeros (for D −→ ∞) is governed by the symplectic dist...

1 9 Ja n 19 99 Low Lying Zeros of Families of L - Functions

Low Lying Zeros of L-functions with Orthogonal Symmetry

We investigate the moments of a smooth counting function of the zeros near the central point of L-functions of weight k cuspidal newforms of prime level N . We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in (− 1 n , 1 n ), as N → ∞ the first n centered moments are Gaussian. By extending the support to (− 1 n−1 , 1 n−1 ), we...

Low-lying Zeros of Number Field L-functions

One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theor...

Low-lying Zeros of L-functions and Random Matrix Theory

By looking at the average behavior (n-level density) of the low-lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups.