Abstract

In this paper we consider the constant term ϕ_K(y, s) of the non-normalized Eisenstein series attached to PSL(2, _K), where K is either ℚ or an imaginary quadratic held of class number one. The main purpose of this paper is to show that for every a ≧ 1 the zeros of the Dirichlet series ϕ_K(a, s) admit a spectral interpretation in terms of eigenvalues of a natural self-adjoint operator Δ_a. This implies that, except for at most two real zeros, all zeros of ϕ_K(a, s) are on the critical line, and all zeros are simple. For K = ℚ this is due to Lagarias and Suzuki [J. C. Lagarias and M. Suzuki, The Riemann hypothesis for certain integrals of Eisenstein series, J. Number Th. 118 (2006), 98–122.] and Ki [H. Ki, Zeros of the constant term of the Chowla-Selberg formula, Acta Arith. 124 (2006), 197–204.].