Simultaneous nonvanishing of the Products of L-functions associated to elliptic cusp forms

Abstract

A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) \neq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, \frac{k-1}{2} < Re(s) < \frac{k}{2}-\epsilon, \frac{k }{2}+\epsilon < Re(s) < \frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $\epsilon.$ This paper concerns the non-vanishing of the product $L^*(f,s)L^*(f,w)$ $(s,w\in \mathbb{C})$ on average.