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 $\mathbf{ N}$ which are abelian extensions of some totally real subfield $\mathbf{ L}$. We note that the smaller the degree of $\mathbf{ L}$ the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing $\mathbf{ L} =\mathbf{ N}^+$ (the maximal totally real subfield of $\mathbf{ N}$) we can choose $\mathbf{ L}$ 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.