Weighted low-lying zeros of L-functionsattached to Siegel modular forms

We study weighted low-lying zeros of spinor and standard L-functions attached to degree 2 Siegel modular forms.We show that 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 .We then show that 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 nonvanishing of central values of those L-functions.