Weighted low-lying zeros of L-functions attached to Siegel modular forms

In this paper, we study weighted low-lying zeros of spinor and standard L-functions attached to degree 2 Siegel modular forms. We show the symmetry type of weighted low-lying zeros of spinor L-functions is symplectic, for test functions whose Fourier transform have support in $(-1,1)$, extending the previous range (-4/15,4/15) by E. Kowalski, A. Saha and J. Tsimerman . We then show the symmetry type of weighted low-lying zeros of standard L-functions is also symplectic. We further extend the range of support by performing an average over weight. As an application, we discuss non-vanishing of central values of those L-functions.