It is well known that Hamburger's converse theorem for the Riemann zeta function does not extend to Dirichlet L-functions, since in this case the functional equations have (in general) many linear independent solutions. Here we characterize the conductors q for which every Dirichlet L-function associated with a primitive character (mod q) is determined, in a general class of Euler products, by its functional equation. Thus, for such conductors we obtain an extension of Hamburger's theorem.
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