Low lying zeros of Artin $$L$$ 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.