Given a CM type \Phi, Colmez assigns two invariants to it:the Faltings height (of the CM abeilian varieties of this CM type) and log derivative of some mysterious virtual Artin representations associated \Phi. He then conjectured that they are equal, and proved the case when the CM field associated to \Phi is abelian. It is a generalization of the well-known Chowla-Selberg formula. The average version of the conjecture was proved recently by two groups of co-authors independently (Yuan-Zhang, Andretta-Goren-Howard-Madapusi-Pera).Some special cases of non-abelian cases were also proved by myself (quartic case), and Bruinier-Howard-Kudla-Rapoport-Yang. In all these non abelian cases and average case, the L-function showed up is very simple---some quadratic Hecke character of some totally real number field. What is the general situation? In this talk, we will describe the interesting case where the L-functions are Hecke characters of some real quadratic fields---seemingly next non-trivial case to crack. This is a joint work with Hongbo Yin.