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