On the mean square of standard \(L\)-functions attached to Ikeda lifts (Q851014)

Let \(f\) be a holomorphic normalized Hecke-eigen cusp form of weight \(2\kappa\) with respect to the full modular group \(\text{SL}(2, \mathbb Z)\), and \(\nu\) be a positive integer with \(\nu\equiv \kappa\pmod 2\). Then \textit{T. Ikeda} [Ann. Math. (2) 154, No. 3, 641--681 (2001; Zbl 0998.11023)] proved the existence of a Hecke-eigen Siegel cusp form \(F_0\) of weight \(\kappa + \nu\) with respect to the Siegel modular group \(\text{Sp}(2\nu, \mathbb Z)\). This Siegel cusp form \(F_0\) is called the Ikeda lift of \(f\), and the associated standard \(L\)-function \(L(s, F_0, st)\) can be decomposed as \[ L(s, F_0, st) = \zeta(s)\prod_{j=1}^{2\nu} L(s + \kappa + \nu - j, f ), \tag{1} \] where \(\zeta(s)\) denotes the Riemann zeta-function and \(L(s, f ) =\sum_{n=1}^\infty a(n)n^{-s}\) is the Hecke L-function associated with \(f\). The first author [Proc. Lond. Math. Soc. (3) 90, No. 2, 297--320 (2005; Zbl 1073.11033)] proved rather sharp estimates of the mean square of \(L(s, F_0, st)\). Let \(I (\sigma; T ) = \int_1^T | L(\sigma + it, F_0, st)|^2\,dt\) for \(T \geq 2\). The ``critical strip'' for \(L(s, F_0, st)\) is \(-\nu + 1/2\leq \sigma\leq \nu + 1/2\). Moreover, the study of \(I (\sigma; T )\) for \(-\nu + 1/2\leq \sigma\leq 1/2\) can be reduced to the study of the case \(1/2 < \sigma\leq\nu + 1/2\) by the functional equation. In this paper the authors prove the following estimates: Theorem 1: For \(\nu + 1\leq \ell\leq 2\nu\), we have \[ I (-\nu + \ell + 1/2; T) \asymp T^{2(2\nu-\ell)^2+1}. \] Theorem 2: We have \[ T^{2\nu(\nu-1)+1} \log T (\log \log T)^{-2} \ll I(1; T)\ll T^{2\nu(\nu-1)+1} \log T (\log \log T)^2. \] Theorem 3: We have \[ T^{2\nu^2+1} \log T (\log \log T)^{-8} \ll I(1/2; T) \ll T^{2\nu^2+1} \log T (\log \log T)^8. \] They first prove Theorem 1, which is a simple supplement of the first author (loc.cit.). The main body of the present paper is the proof of Theorems 2 and 3, for which they use a method of \textit{K. Ramachandra} and the second author [Acta Arith. 109, No. 4, 349--357 (2003; Zbl 1036.11045)]. Finally they discuss further examples (Theorems 4, 5 and 6), including the case of spinor \(L\)-functions attached to Saito-Kurokawa lifts.