A spectral interpretation of the zeros of the constant term of certain Eisenstein series

Abstract

In this paper we consider the constant term $\phi_K(y,s)$ of the non-normalized Eisenstein series attached to $\PSL(2,\cO_K)$, where $K$ is either $\Q$ or an imaginary quadratic field of class number one. The main purpose of this paper is to show that for every $a\ge 1$ the zeros of the Dirichlet series $\phi_K(a,s)$ admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator $\Delta_a$. This implies that, except for at most two real zeros, all zeros of $\phi_K(a,s)$ are on the critical line, and all zeros are simple. For $K=\Q$ this is due to Lagarias and Suzuki and Ki.