Weighted first moments of some special quadratic Dirichlet LL-functions

In this paper, we obtain asymptotic formulas for weighted first moments of central values of families of primitive quadratic Dirichlet $L$-functions whose conductors comprise only primes that split in a given quadratic number field. We then deduce a non-vanishing result of these $L$-functions at the point $s=1/2$.Comment: 7 page

Simultaneous non vanishing of GL(3)GL(3) LL-functions

The main objective of this article is to compute a first moment for product of Dirichlet and twisted self-dual $GL(3)$ $L$-functions. We discuss the possible simultaneous non vanishing at the central point. We use properties of symmetric squares $L$-functions.Comment: The result is conditioned by a false hypothesi

A note on exceptional characters and non-vanishing of Dirichlet LL-functions

We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $\psi$ is a real primitive character modulo $D \in \mathbb{N}$ with $L(1, \psi) \ll (\log D)^{-25-\varepsilon}$, then, for any prime $q \in [D^{300}, D^{O(1)}]$, one has $L(1/2, \chi) \neq 0$ for almost all Dirichlet characters $\chi \pmod{q}$.Comment: Published version, incorporated referee's comment

Mollified Moments of Cubic Dirichlet L-Functions over the Eisenstein Field

We prove, assuming the generalized Riemann Hypothesis (GRH) that the density of the $L$-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point $s=1/2$ is positive. This is achieved by computing the first mollified moment assuming a subconvexity bound, and obtaining a sharp upper bound for the higher mollified moments for these $L$-functions under GRH. The proportion of non-vanishing is explicit, but extremely small.Comment: 46 page

Non-vanishing of twists of GL4(AQ)\text{GL}_4(\mathbb{A}_{\mathbb{Q}}) LL-functions

Let $\pi$ be a unitary cuspidal automorphic representation of $\text{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$. Let $f \geq 1$ be given. We show that there exists infinitely many primitive even (resp. odd) Dirichlet characters $\chi$ with conductor co-prime to $f$ such that $L(s, \pi \otimes \chi)$ is non-vanishing at the central point. Our result has applications for the construction of $p$-adic $L$-functions for $\text{GSp}_{4}$ following Loeffler-Pilloni-Skinner-Zerbes, the Bloch-Kato conjecture and the Birch-Swinnerton-Dyer conjecture for abelian surfaces following Loeffler-Zerbes, strong multiplicity one results for paramodular cuspidal representations of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and the rationality of the central values of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ $L$-functions in the remaining non-regular weight case.Comment: 45 page

L-functions with n-th order twists

Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s,\chi) for n-th order Hecke characters: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.Comment: 21 pages, 1 figur