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Katz showed that the L-functions of all Dirichlet characters of F_q(t), withconductor a fixed power of a degree one prime, are equidistributed in the limitas q goes to infinity. We generalize this statement to the L-functions oftwists of an arbitrary Galois representation by Dirichlet characters, includingindependence of the L-functions of twists of different representations by thesame Dirichlet character. A similar generalization, without the independencestatement, for characters with squarefree conductor, was proven by Hall,Keating, and Roditty-Gershon.

The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fields