Universality of Hecke $L$-functions in the Grossencharacter-aspect

Number Theory 19 Statistics for low - lying zeros of Hecke L - functions in the level aspect

We would like to provide evidence for the fact that zeros of L-functions seem to behave statistically as eigenvalues of random matrices of large rank throughout the instance of Hecke L-functions. First, we remind you of Iwaniec-Luo-Sarnak’s results on one-level densities for low-lying zeros of Hecke L-functions (see [5]) and Katz-Sarnak’s results on one-level densities for eigenvalues of orthog...

(Almost) primitivity of Hecke L-functions

Let f be a cusp form of the Hecke space M0(λ, k, ) and let Lf be the normalized L-function associated to f . Recently it has been proved that Lf belongs to an axiomatically defined class of functions S̄. We prove that when λ ≤ 2, Lf is always almost primitive, i.e., that if Lf is written as product of functions in S̄, then one factor, at least, has degree zeros and hence is a Dirichlet polynomial...

Effective Nonvanishing of Canonical Hecke L-functions

Motivated by work of Gross, Rohrlich, and more recently Kim, Masri, and Yang, we investigate the nonvanishing of central values of L-functions of “canonical” weight 2k−1 Hecke characters for Q( √ −p), where 3 < p ≡ 3 (mod 4) is prime. Using the work of Rodriguez-Villegas and Zagier, we show that there are nonvanishing central values provided that p ≥ 6.5(k−1) and (−1) ( 2 p ) = 1. Moreover, we ...

An Explicit Formula for Hecke L-functions

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp forms of weight k for Hecke congruence subgroups, lie on the critical line.

The Second Moment of Dirichlet Twists of Hecke L-functions