Ratios conjecture for primitive quadratic Hecke $L$-functions

Abstract

We develop the ratios conjecture with one shift in the numerator and denominator in certain ranges for families of primitive quadratic Hecke $L$-functions of imaginary quadratic number fields with class number one using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain unconditional asymptotic formulas for the first moments of central values of these families of $L$-functions with error terms of size that is the square root of that of the primary main terms.