Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect

Statistics for low-lying zeros of symmetric power L-functions in the level aspect

We study one-level and two-level densities for low lying zeros of symmetric power L-functions in the level aspect. It allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.

On the value-distribution of the logarithms of symmetric square L-functions in the level aspect

We consider the value distribution of logarithms of symmetric square L-functions associated with newforms of even weight and prime power level at real s> 1/2. We prove that certain averages of those values can be written as integrals involving a density function which is related with the Sato-Tate measure. Moreover, we discuss the case of symmetric power L-functions.

对称平方$L$-函数在权重方面的非消失性

Non-vanishing of symmetric square $L$-functions in the weight aspect

本文通过证明对称平方$L$-函数第二矩的新渐近公式，将全纯Hecke尖点形式在中心点非消失的比例从70.37%提升至76.69%。

On the value-distribution of the logarithms of symmetric power L-functions in the level aspect

We consider the value distribution of logarithms of symmetric power L-functions associated with newforms of even weight and prime power level. In the symmetric square case, under certain plausible analytical conditions, we prove that certain averages of those values in the level aspect, involving continuous bounded or Riemann integrable test functions, can be written as integrals involving a density function (the "M-function") which is related with the Sato-Tate measure. Moreover, even in the case of general symmetric power L-functions, we show the same type of formula when for some special ty…

Subconvexity for Symmetric Square L-functions off-centre

Let $p$ be a prime. Let $f$ be a holomorphic modular form of level $p$ with trivial nebentypus. We prove the bound $L\left(\text{sym}^2f, \frac{1}{2} + it\right) \ll_{f,ε} p^{1/2+ε}t^{3/4-1/12 + ε}$. This bound is subconvex in the $t$-aspect and almost convex in the level aspect simultaneously.

The first moment of central values of symmetric square $L$-functions in the weight aspect

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of $L(\frac{1}{2},{\rm{sym}}^2f)$ for $f\in\mathcal{H}_k$ as even $k\rightarrow\infty$, where $\mathcal{H}_k$ is an orthogonal basis of weight-$k$ Hecke eigencuspforms for $SL(2,\mathbb{Z})$. The approach taken in this note allows us to extract two secondary main terms from the error term $O(k^{-\frac{1}{2}})$ in previous studies. More interestingly, our resul…