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We show that the central value of class group L-functions of CM fields can beexpressed in terms of derivatives of real-analytic Hilbert Eisenstein series atCM points. Then, following an idea of Iwaniec and Kowalski we obtain aconditional explicit lower bound of class numbers of CM fields under a weakerassumption. Some results in the proof lead to an effective nonvanishing resultfor class group L-functions of general CM fields, generalizing the only knownineffective results.

An Explicit CM Type Norm Formula and Effective Nonvanishing of Class Group L-functions for CM Fields