Square root formulas for central values of Hecke \(L\)-series. II
The results proved in Part I [the author and \textit{D. Zagier}, Proc. 3rd Conf. Can. Number Theory Assoc., Kingston, Canada 1991, 81-99 (1993; Zbl 0791.11060)] are extended to imaginary quadratic fields with arbitrary odd discriminant \(-d<-3\). More precisely, let \(L(\psi, s)\) be the \(L\)-series attached to a Hecke character \(\psi\) of \(K\) of ``weight \(2k-1\)'', i.e. the associated modular form has weight \(2k\). Then it is shown that to \(\psi\) one can attach a well-defined genus \(G_ \psi\) of \(K\) such that the central value \(L(\psi, k)\) up to a simple normalizing factor is given as the square of the absolute value of a certain linear combination of values of the \((k- 1)\)-st non-holomorphic derivative of a theta series of weight \(k+ 1/2\) at CM points corresponding to ideals in \(G_ \psi\).
non- holomorphic derivative of a theta series
scientific article; zbMATH DE number 558969