On a Zeta Function Associated to a Quadratic Order

Abstract

For a positive integer N, we derive a Kronecker type limit formula for a Dedekind type zeta function $\zeta _{\mathcal {O}_{K}(N)}(s;[\mathfrak {a}])$ associated to a wide ideal class of a quadratic order of conductor N of a quadratic field K. As consequences, we establish a Chowla–Selberg type formula for a modular form for $\Gamma _{0}(N)$, give a proof to a classical class number relation of a quadratic field, and find an asymptotic formula for the average value of the number of representations by a primitive positive definite quadratic form of discriminant $N^{2}d$ with d fundamental. Furthermore, as an application, we obtain a formula for the Petersson norm of a theta function attached to a class character of the ideal group of conductor N of $\mathcal {O}_{K}$.