On the Values at Negative Half-Integers of the Dedekind Zeta Function of a Real Quadratic Field

Abstract

The zeta function $\zeta (A,s)$ associated with a narrow ideal class $A$ for a real quadratic field can be decomposed into $\sum \nolimits _Q {{Z_Q}(s)}$, where ${Z_Q}(s)$ is a Dirichlet series associated with a quadratic form $Q(x,y) = a{x^2} + bxy + c{y^2}$, and the summation is over finite reduced quadratic forms associated to the narrow ideal class $A$. The values of ${Z_Q}(s)$ at nonpositive integers were obtained by Zagier [16] and Shintani [12] via different methods. In this paper, we shall obtain the values of ${Z_Q}(s)$ at negative half-integers $s = - 1/2, - 3/2, \ldots , - m + 1/2, \ldots$. The values of ${Z_Q}(s)$ at nonpositive integers were also obtained by our method, and our results are consistent with those given in [16].