Computation of relative class numbers of CM-fields by using Hecke L-functions

We develop an efficient technique for computing values at s = 1 of Hecke L-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing L = N+ (the maximal totally real subfield of N) we can choose L real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.