Generalizations of Hamburger's Theorem

Question

(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).

An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet series satisfying Riemann's functional equation" (Knopp took the statement from a work of Hecke), asserts that the Riemann zeta function $\zeta$ is determined by its functional equation, in the following sense: suppose $R(s) = \pi^{-s} \Gamma(s) \varphi(s)$, where $\varphi(s)$ is a meromorphic function. If $\varphi(s)$ satisfies:

1. $P(s) \varphi(s)$ is entire for some polynomial $P$;
2. $\varphi(s)$ satisfies the functional equation $R(s) = R(1/2 - s)$; and
3. a) Both $\varphi(s)$ and $\varphi(s/2)$ can be expressed as a Dirichlet series convergent somewhere, so that $\varphi(s) = \sum_{n \geq 1} b(n) n^{-2s}$ b) $\varphi(s)$ can be expressed as a Dirichlet series, and the only pole of $\varphi$ is at $s = 1/2$.

Then both 1, 2, 3a) and 1,2, 3b imply that $\varphi$ is a scalar multiple of $\zeta(2s)$.

Has Hamburger's theorem been generalized to other functional equations appearing for well-known $L$-functions, such as Dirichlet $L$-functions?

Answer 1

Hamburger's theorem has been generalized in various ways to automorphic $L$-functions (of arbitrary degree). Such generalizations are called "converse theorems", and they play a central role in the Langlands program. See for example this paper by Cogdell and Piatetski-Shapiro, especially the historical notes on the first two pages. See also these notes by Cogdell.