On calculation of \(L_ K(1,\chi)\) for some Hecke characters (Q1323266)

Let \(K\) be a quadratic extension of a totally real number field; given an integral ideal \({\mathfrak f}\) and a subset \(\Omega\) of the set of real places of \(K\), let \(H({\mathfrak f}\Omega)\) be the ray class group of conductor \(({\mathfrak f}, \Omega)\). For \(c\in H({\mathfrak f}\Omega)\), let \(\zeta(s,C)\) denote the corresponding partial zeta-function; write \[ \zeta(s,C)= g({\mathfrak f} \Omega) (R(K)/ W(K)) s^{\ell-1}+ \kappa(K,C) s^ \ell+ O(s^{\ell+1}), \] as \(s\to 0\), where \(\ell= r_ 1+ r_ 2- e_ 1\), \(e_ 1:= \text{card } \Omega\), \(g({\mathfrak f} \Omega)=1\) if \({\mathfrak f}= {\mathfrak o}\), \(\Omega=\emptyset\) and \(g({\mathfrak f} \Omega)=0\) if \(({\mathfrak f}, \Omega)\neq ({\mathfrak o},\emptyset)\). Here \({\mathfrak o}\) denotes the ring of integers of \(K\), and \(r_ 1\) (resp. \(r_ 2\)) is equal to the number of real (resp. complex) places of \(K\); as usual, \(R(K)\) and \(W(K)\) stand for the regulator of \(K\) and the number of roots of unity in \(K\) respectively. The author expresses the first non-trivial coefficient in the Taylor expansion of \(\zeta(s,C)\) at \(s=0\) in terms of certain ``periods'' associated to arithmetic subgroups of Hilbert modular groups. His formulae can be conveniently used to compute \(\kappa(K,C)\), and the author concludes his paper with a few examples of such computations. The expression for \(\kappa(K,C)\) takes especially simple form if \(K\) has exactly two real places (that is, if \(r_ 1=2\)).