Lower bound for the higher moment of symmetric square \(L\)-functions. (English) Zbl 1417.11075

Ji, Guanghua

Rocky Mt. J. Math. 46, No. 3, 915-923 (2016).

Summary: Let \(\mathcal{S}_k(N)\) be the space of holomorphic cusp forms of weight \(k\), level \(N\) and let \(\mathcal{B}_k(N)\) be an orthogonal basis of \(\mathcal{S}_k(N)\) consisting of newforms. Let \(L(s,\,\text{sym}^2 f)\) be the symmetric square \(L\)-function of \(f\in \mathcal{B}_k(N)\). In this paper, the lower bound of the higher moment of \(L(1/2,\,\text{sym}^2 f)\) is established, i.e., for any even positive number \(r\), \[ \sum_{f\in\mathcal{B}_{k}(N)}\omega_f^{-1}L\bigg(\frac{1}{2},\,\text{sym}^2 f\bigg)^r \gg (\log N)^{r(r+1)/2} \] holds for \(N\to \infty \).

MSC:

11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F11	Holomorphic modular forms of integral weight

Keywords:

holomorphic forms; lower bound; symmetric square \(L\)-functions

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