Lower order terms for the one-level density of a symplectic family of Hecke L-functions

In this paper we apply the $L$-function Ratios Conjecture to compute one-level density for a symplectic family of $L$-functions attached Hecke characters infinite order. When support Fourier transform corresponding test function $f$ reaches $1$, observe transition in main term, as well lower order term. The term is line with behavior recently observed by D. Fiorilli, J. Parks, and A. S\"odergren their study quadratic Dirichlet $L$-functions. We then directly calculate terms functions such that supp($\widehat{f}) \subset [-\alpha,\alpha]$ some $\alpha <1$, unconditional result agreement prediction provided Conjecture. As analytic conductor these L-functions grow twice large (on logarithmic scale) cardinality question, optimal can be expected current methods. Finally corollary deduce that, under GRH, at least 75$\%$ do not vanish central point.