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There are in fact at least two axiomatic characterizations of $\zeta(s).$ On 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

produces infinitely many linearly independent dirichlet series!

Sources:
http://www.rowan.edu/open/depts/math/HASSEN/Papers/paper1.pdf
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There are in fact at least two axiomatic characterizations of $\zeta(s).$ On of them is given by Hecke and one of them was given by Hamburger.Both of them have been given already by the other answerers.

What I can add though

Hamburger's Theorem states: Suppose $D(s)$ is that both descriptions are distinct a dirichlet series, convergent in the sense some half plane and whose coefficients have polynomial growth. If $$$$1) There exists a polynomial $P(s)$ so that you cannot take $P(s)D(s)$ is entire and of finite order.

2) $D(s/2)$ is also dirichlet series. That is, the weaker axiom from each descriptioncoefficients of $D(s)$ are supported on squares.

3) If you do this you end up with infinitely many linearly independent Dirichlet $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ 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. This If $$$$1) $(s-1/2)D(s)$ is the so-called failure 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" theorem. Below I think where$$$$1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is paper where all entire and of this is discussedfinite order.

Knopp, M

2) The coefficients of $D(s)$ have arbetrary support.Hamburger's theorem on zeta(s

3) and the abundance principle for Dirichlet If $\xi(s)= \pi^{-s}\Gamma(s)D(s)$ then $\xi(1/2-s)=\xi(s).$

Produces infinitely many linearly independent dirichlet serieswith functional equations (invited paper), in Number Theory, Indian National Science Academy, New Delhi, eds.!

Sources:R.P. Bambah et al. (2000) 201-216.
http://www.rowan.edu/open/depts/math/HASSEN/Papers/paper1.pdf
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There are in fact at least two axiomatic characterizations of $\zeta(s).$ On of them is given by Hecke and one of them was given by Hamburger. Both of them have been given already by the other answerers.

What I can add though is that both descriptions are distinct in the sense that you cannot take the weaker axiom from each description. If you do this you end up with infinitely many linearly independent Dirichlet series. This is the so-called failure of "Big Mac" theorem. Below I think is paper where all of this is discussed.

Knopp, M. Hamburger's theorem on zeta(s) and the abundance principle for Dirichlet series with functional equations (invited paper), in Number Theory, Indian National Science Academy, New Delhi, eds.: R.P. Bambah et al. (2000) 201-216.