Related papers: Non-vanishing for cubic $L$--functions

Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that…

In this paper, we study moments of central values of cubic Hecke $L$-functions in $\mathbb{Q}(i)$, and establish quantitative non-vanishing result for those values.

We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not…

With the method of moments and the mollification method, we study the central $L$-values of GL(2) Maass forms of weight $0$ and level $1$ and establish a positive-proportional nonvanishing result of such values in the aspect of large…

We obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over F_q[t]. We solve this problem in the non-Kummer setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs rely on…

We prove that there are infinitely many Maass--Hecke cuspforms over the field $\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series does not vanish at the center of the critical strip. This is done by using a result…

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result…

A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed…

In this paper, we show that, on average, the derivatives of $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$,…

We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the…

We show that for at least 3/8 of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(1/2,\chi)$ does not vanish.

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose…

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues…

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of…

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least $35.9\%$ of the primitive Dirichlet $L$-functions.

We obtain an asymptotic formula for all moments of Dirichlet $L$-functions $L(1,\chi)$ modulo $p$ when averaged over a subgroup of characters $\chi$ of size $(p-1)/d$ with $\varphi(d)=o(\log p)$. Assuming the infinitude of Mersenne primes,…

We investigate the consequences of natural conjectures of Montgomery type on the non-vanishing of Dirichlet $L$-functions at the central point. We first justify these conjectures using probabilistic arguments. We then show using a result of…

In this paper, we prove that if the Fourier coefficients of a $\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maa\ss\ cusp form $\pi$ are not too correlated with additive characters, then there exists infinitely many Dirichlet characters such that…

For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then…

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke…