  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 $(n-r, r)$'.
