A peculiar modular form of weight one

Abstract

In this paper we construct a modular form f of weight one attached to an imaginary quadratic field K. This form, which is non-holomorphic and not a cusp form, has several curious properties. Its negative Fourier coefficients are non-zero precisely for neqative integers -n such that n >0 is a norm from K, and these coefficients involve the exponential integral. The Mellin transform of f has a simple expression in terms of the Dedekind zeta function of K and the difference of the logarithmic derivatives of Riemann zeta function and of the Dirichlet L-series of K. Finally, the positive Fourier coefficients of f are connected with the theory of complex multiplication and arise in the work of Gross and Zagier on singular moduli.