Value-distribution of quartic Hecke $L$-functions

Abstract:

Let cc be a square-free Gaussian integer such that cc is congruent to 1 modulo 16. For fixed real σ>1/2σ>1/2, we show that there is an asymptotic distribution function FσFσ for the values of logarithm of the product of LL-functions: Lc(s)=L(s,χc)L(s,χc¯¯¯¯¯)Lc(s)=L(s,χc)L(s,χc¯) as cc varies, where L(s,χc)L(s,χc) is the Hecke LL-function associated with the quartic residue character modulo c. Moreover, we express the characteristic function of FσFσ explicitly as a product indexed by the prime ideals in the ring of Gaussian integers. This is a joint work with L. Zhao.