Number Theory 19 Statistics for low - lying zeros of Hecke L - functions in the level aspect

We would like to provide evidence for the fact that zeros of L-functions seem to behave statistically as eigenvalues of random matrices of large rank throughout the instance of Hecke L-functions. First, we remind you of Iwaniec-Luo-Sarnak’s results on one-level densities for low-lying zeros of Hecke L-functions (see [5]) and Katz-Sarnak’s results on one-level densities for eigenvalues of orthogonal random matrices (see [6]). Then, we explain that Hughes and Miller (see [1]) found a new example of a very strange phenomenon discovered by Hughes and Rudnick (see [2]) called mock-Gaussian behavior. These works were carried on by the author and Royer in the context of low-lying zeros of symmetric power L-functions in the level aspect (see [7]).