The low-lying zeros of L-functions associated to non-Galois cubic fields

Abstract: We study the low-lying zeros of Artin L-functions associated to non-Galois cubic number fields through their one- and two-level densities. In particular, we find new precise estimates for the two-level density with a power-saving error term. We apply the L-functions Ratios Conjecture to study these densities for a larger class of test functions than unconditional computations allow. By reviewing a known Ratios Conjecture prediction, due to Cho, Fiorilli, Lee, and Södergren, for the one-level density, we isolate a phase transition in the lower-order terms, which reveals a striking symmetry. Our computations show that the same symmetry exists in the one-level density of several other families, that have previously been studied in the literature, and this motivates us to formulate a conjecture extending one part of the Katz–Sarnak prediction for families of symplectic symmetry type. Moreover, we isolate several phase transitions in the lower-order terms of the two-level density. To the best of our knowledge, this is the first time such phase transitions have been observed in any n-level density with n ≥ 2.