Zeta functions for ideal classes in real quadratic fields, at s=0s=0

Let K be a real quadratic field with discriminant d, and for a (fractional) ideal a of K, let Na be the norm of a. For a given fractional ideal I of K, and Dirichlet character χ of conductor q, we defineζI(s,χ)=ζCl(I)(s,χ):=∑aχ(Na)(Na)s where the sum is over all integral ideals a of K which are equivalent to I  . We give a short, easily computable formula to evaluate ζI(0,χ)ζI(0,χ), using familiar objects from considerations of K  . We generalize our formula to ζI(1−k,χ)ζI(1−k,χ) with k⩾1k⩾1, though the result obtained is not quite so satisfactory as that for k=1k=1. We discuss connections between these formulae and small class numbers.