Ratios conjecture of cubic $L$-functions of prime moduli

Abstract

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the generalized Riemann hypothesis. As applications, we evaluate asymptotically the first moment of central values as well as the one-level density of the same family of $L$-functions.

14 pages, revised with accordance to the referee's comments and suggestions

References in corpus (4)

• Ratios conjecture for quadratic Hecke $L$-functions in the Gaussian field
• Ratios conjecture for primitive quadratic Hecke $L$-functions
• Notes on zeta ratio stabilization
• Ratios conjecture for quadratic twist of modular $L$-functions