For all sufficiently large primes q, the lowest non-trivial zero of Dirichlet L-functions modulo q has height below 1075 times the average spacing, with a positive proportion having even smaller first zeros.
Average non-vanishing of DirichletL-functions at the central point
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Under GRH, four twisted Dirichlet L-functions at 1/2 + it are simultaneously non-zero for a positive proportion of characters χ mod q (q prime, sufficiently large); unconditionally, this holds for infinitely many χ but with proportion tending to zero.
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Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
For all sufficiently large primes q, the lowest non-trivial zero of Dirichlet L-functions modulo q has height below 1075 times the average spacing, with a positive proportion having even smaller first zeros.
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Simultaneous non-vanishing of Dirichlet L-functions
Under GRH, four twisted Dirichlet L-functions at 1/2 + it are simultaneously non-zero for a positive proportion of characters χ mod q (q prime, sufficiently large); unconditionally, this holds for infinitely many χ but with proportion tending to zero.