TITLE: Axioms for Riemann $\zeta$ function
QUESTION [9 upvotes]: Are there any set of axioms that completely characterize the Riemann zeta function?
 i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.

REPLY [5 votes]: There are in fact at least two axiomatic characterizations of $\zeta(s).$  One of them is given by Hecke and one of them was given by Hamburger.
Hamburger's Theorem states:  Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth.  If $$$$
1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order.
2) $D(s/2)$ is also dirichlet series. That is, the coefficients of $D(s)$ are supported on squares.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
Then $D(s)=C\zeta(2s).$
Hecke's version states: Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth.  If $$$$
1) $(s-1/2)D(s)$ is entire and of finite order. 
2) The coefficients of $D(s)$ have arbetrary support.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
Then $D(s)=C\zeta(2s).$
A little extra information: These theorems cannot be combined. The so-called "Big Mac Theorem" where$$$$
1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order.
2) The coefficients of $D(s)$ have arbetrary support.
3) If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$
produces infinitely many linearly independent dirichlet series!
Sources:
http://www.rowan.edu/open/depts/math/HASSEN/Papers/paper1.pdf