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 see non-Gaussian behavior; in particular the odd centered moments are non-zero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in (− 2 n , 2 n ) if 2k ≥ n. The n centered moments agree with Random Matrix Theory in this extended range, providing additional support for the Katz-Sarnak conjectures. The proof requires calculating multidimensional integrals of the non-diagonal terms in the Bessel-Kloosterman expansion of the Petersson formula. We convert these multidimensional integrals to one-dimensional integrals already considered in the work of Iwaniec-LuoSarnak, and derive a new and more tractable expression for the n centered moments for such test functions. This new formula facilitates comparisons between number theory and random matrix theory for test functions supported in (− 1 n−1 , 1 n−1 ) by simplifying the combinatorial arguments. As an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point.