Source:
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (22):6955-6978; 10.1093/imrn/rnv320 2016

Abstract:
A well-known result of Iwaniec and Sarnak states that for at least one-third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of non-vanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime power conductor.