A mean value theorem for the Dedekind zeta-function of a quadratic number field

Let \(K = \mathbb{Q}(\sqrt d )\) be any quadratic number field with discriminantd. ζ _K (s) denotes the Dedekind zeta-function. The purpose of this note is to prove the following asymptotic formula: $$\int\limits_0^T {|\zeta _K ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + it)|^2 dt = ({6 \mathord{\left/ {\vphantom {6 {\pi ^2 }}} \right. \kern-\nulldelimiterspace} {\pi ^2 }})} \prod\limits_{p/d} {(1 + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})^{ - 1} \cdot R_K^2 \cdot T \cdot \log ^2 T + O_\varepsilon \left\{ {\left| d \right|1 + \varepsilon \cdot T \cdot \log T} \right\},} $$ where the implied constant depends only on ε. HereR _K, denotes the residue of ζ _K (s) ats=1.