CM fields of Dyhedral type and the Colmez conjecture

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin $L$-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the `abelian case of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of $Q$ (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for `unitary CM types of signature $(n-1, 1)$ and holds on average for `unitary CM types of a fixed CM number field of signature $(n-r, r)$.