In 1921 Hamburger proved that Riemann's functional equation characterizes the Riemann zeta function in the space of functions representable by ordinary Dirichlet series satisfying certain regularity conditions. We consider solutions to a more general functional equation with real weight k. In the case of Hamburger's theorem, k = $$ - \frac{1}{2}$$ . We show that, under suitable conditions, the generalized functional equation admits no nontrivial solutions for k > 0 unless k = $$ - \frac{1}{2}$$ . Our proof generalizes an elegant proof of Hamburger's theorem given by Siegel, and employs a generalized integral transform.