Low lying zeros of Artin $L$-functions

Abstract

We study the one-level density of Artin $L$-functions twisted by a cuspidal automorphic representation under the strong Artin conjecture and certain conjectures on counting number fields. Our result is unconditional for $S_3$-fields. For a non-self dual $\pi$, it agrees with the unitary type $\text {U}$. For a self-dual $\pi$ whose symmetric square $L$-function $L(s,\pi ,\text {Sym}^2)$ has a pole at $s=1$, it agrees with the symplectic type $\text {Sp}$. For a self-dual $\pi$ whose exterior square $L$-function $L(s,\pi ,\wedge ^2)$ has a pole at $s=1$, the possible symmetry types are $\text {O}$, $\text {SO(even)}$, or $\text {SO(odd)}$. When $\pi =1$, for $S_3$ cubic fields and $S_4$ quartic fields, we rediscover Yang’s one-level density result in his thesis (Yang 2009). In the last section, we compute the one-level density of several families of Artin $L$-functions arising from parametric polynomials.