In the comments section to Willie Wong's answer, the following Dirichlet series came up: the Riemann $\zeta$-function, Dirichlet $L$-functions, and Ramanujan's series $\sum_{n \geq 1}\tau(n) n^{-s}$, where $\tau(n)$ is the coefficient of $q^n$ in $\Delta(q) = q\prod_{n=1}^{\infty} (1-q^n)^{24}$.
First note that the $\zeta$-function is a special case of a Dirichlet $L$-function (it is the $L$-function of the trivial character).
Now what is it that Dirichlet $L$-functions and Ramanujan's series have in common? Well, they are all automorphic $L$-functions.
An automorphic form (for the group $\mathrm{GL}_n$ over $\mathbb Q$; there are generalizations where $\mathbb Q$ is replaced by an arbitrary number field $F$ and $\mathrm{GL}_n$ is replaced by an arbitrary reductive group, but to simplify the explanations, I will focus just on the simplest level of generality here) is a function on the product $\mathrm{GL}_n(\mathbb R)\times \mathrm{GL}_n(\mathbb Z/N\mathbb Z)$ for some integer $N \geq 1$ which
is
invariant under the natural (diagonal) action of $\mathrm{GL}_n(\mathbb Z)$;
grows moderately at infinity with respect to the $\mathrm{GL}_n(\mathbb R)$-coordinates;
satisfies a suitable differential equation in the $\mathrm{GL}_n(\mathbb R)$-coordinates.
Rather than explaining the generalities in more detail (they can be found in many places), I think it's better to illustrate them:
E.g. Dirichlet characters arise in the case $n = 1$: they are defined as functions
on $(\mathbb Z/N\mathbb Z)^{\times} =: \mathrm{GL}_1(\mathbb Z/N\mathbb Z)$,
and so we can make them into functions on $\mathrm{GL}_1(\mathbb R)\times
\mathrm{GL}_1(\mathbb Z/N\mathbb Z)$ by defining them to be trivial on the
$\mathbb R^{\times}$-coordinate.
E.g. If $f(\tau)$ is a holomorphic modular form of weight $k$ and level one (where $\tau$ is an upper half-plane variable as usual), we can make $f$ into a function on
$\mathrm{GL}_2(\mathbb R)$ by first identifying this matrix group with the collection of bases of $\mathbb R^2$, then identifying $\mathbb R^2$ with $\mathbb C$,
and then defining, for any $\mathbb R$-basis $\omega_1,\omega_2$ of $\mathbb C$,
$f(\omega_1,\omega_2) := \omega_2^{-k} f(\omega_1/\omega_2)$. (This presumes
that $\omega_1/\omega_2$ is in the upper half-plane rather than the lower,
for simplicity.) Thus we get a function of the required kind (with $N = 1$).
The usual modularity condition becomes invariance under $\mathbb GL_2(\mathbb Z)$. The moderate growth condition becomes the condition that the Fourier expansion of $f$ involves only non-negative powers of $e^{2 \pi i \tau}$. The differential equation is the Cauchy--Riemann equation expressing holomorphy of $f$.
Higher level modular forms will involve values of $N$ that are $> 1$.
E.g. Maass forms are similar to the preceding example, except that now the
differential equation expresses that a Maass form is an eigenvector of the Laplacian.
For any fixed $n$ and fixed $N$, we have Hecke operators acting on the space
of automorphic forms, labelled by primes $p$ not dividing $N$, and so we can
consider Hecke eigenforms. In the case of Dirichlet characters, the fact that they are characters of $(\mathbb Z/N\mathbb Z)^{\times}$ (rather than just arbitrary functions) can be reinterpreted as saying that they are Hecke eigenforms.
Of course, Ramanujan's $\Delta$ is well-known to be a Hecke eigenform of weight $12$ and level $1$.
Given an automorphic Hecke eigenform we can use the Hecke eigenvalues to make
an Euler product Dirichlet series, which will give Dirichlet $L$-functions
for Dirichlet characters, and Ramanujan's Dirichlet series for $\Delta$.
(In the Dirichlet character case, if a prime $p$ divides the conductor $N$, we just have a trivial factor in the Euler product for that prime; when $n > 1$,
and $N > 1$, it is a bit more of a battle to figure out what Euler factors
to put in at the primes dividing $p$, but it can be done.)
Actually, it is better to restrict to cuspidal automorphic Hecke eigenforms.
Cuspidal is a vacuous condition when $n = 1$ (i.e. in that case we agree to call
everything cuspidal), and when $n > 1$ we replace "moderate growth at infinity" by "rapid decay at infinity", suitably interpreted. I'll assume that all my eigenforms are cuspidal
form now on. (E.g. $\Delta$ is cuspidal.)
In this way we get a natural class of $L$-functions which have:
meromorphic continuation to the whole complex plane, which is in fact
holomorphic with the sole exception of Riemann's $\zeta$.
Functional equation with completely understood $\Gamma$-factors. E.g. for a
weight $k$ modular form of level one, if the $p$th Hecke eigenvalue is $a_p$,
then the $L$-function is $\prod_p (1 - a_p p^{-s} + p^{k - 1 - 2s})^{-1},$ and the functional equation relates $s$ and $k - s$. For $\Delta,$ I've already noted that $k = 12$. (In general the functional equation relates the $L$-series of an automorphic eigenform with $L$-series of its "complex conjugate" suitably understood, just as in the case of Dirichlet characters that are not necessarily real valued.)
Conjecturally, they should all satisfy the analogue of RH, i.e. all non-trivial zeroes should lie on the critical line, in the centre of the critical strip.
Note incidentally that it is easy to change the apparent form of the functional equation. E.g. if we make a change of variable $s \mapsto s + 11/2$ in Ramanujan's series, then the functional equation will become $s \mapsto 1 -s$
rather than $s \mapsto 12 -s $, and the critical line will be $\Re s = 1/2$, just as in the $\zeta$-function case.
All cuspidal automorphic $L$-functions can be renormalized in a similar way, so that the symmetry of the functional equation is $s \mapsto 1 -s$. This is called unitary normalization, and is common in the automorphic forms literature.
Up to rescaling, there are only countably many automorphic eigenforms altogether (just because if we fix the level $N$ and (appropriately generalized version of) the weight the space of automorphic forms is finite dimensional) and so altogether we are talking about a very special class of just countably many Dirichlet series, but these seem to be the ones that naturally generalize $\zeta(s)$.
By the way, this general point of view is due to Langlands, and forms a part of the general Langlands program.
Another point of view was given by Selberg, which focuses more on capturing the analytic properties necessary for getting good properties of a Dirichlet series, rather than beginning from a conceptual construction (as in the automorphic point of view). Namely, he introduced the Selberg class of Dirichlet series. Note that part of his axioms are an Euler product, analytic continuation, and functional equation.
My sense is, though, that people expect the Selberg class of Dirichlet series to more-or-less coincide with the class of automorphic $L$-functions, so I think it is just two points of view on the same question: Langlands is showing how to construct "good" Dirichlet series, and Selberg is writing down the properties a "good" Dirichlet series should satisfy. It turns out that "good" Dirichlet series are so special, though, that however you try to pick them out, you seem to end up with the same very special collection, namely the automorphic ones.
