Under GRH, in every Galois orbit modulo p^k, a positive proportion of characters chi satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0 for distinct imprimitive twists eta1, eta2.
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Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits
Under GRH, in every Galois orbit modulo p^k, a positive proportion of characters chi satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0 for distinct imprimitive twists eta1, eta2.