The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_{m =1}^\infty \frac{1}{(m^2)^s} = \sum_{m=1}^\infty \frac{1}{m^{2s}} = \zeta(2s)$, which stands roughly in the same relation to $\theta(z)$ as the $L$-function of a modular form $f$ stands to $f$. (They are both essentially Mellin transforms).

(Of course, there are a few issues here - $\theta$ is only a twisted modular form, the Fourier expansion for $\theta$ is in terms of $e^{\pi i z}$ instead of $e^{2\pi i z}$, there is the factor of $2$ appearing in the argument, etc.)

In the case of a weight $2k$ modular form $f$ (at least in the case that $f$ is a Hecke eigenform), there is an associated Galois representation with the same $L$-function by the Eichler-Shimura construction, and the Langlands conjectures for $GL_2(\mathbb{Q})$ predict that every Galois representation of a certain type arises this way.

Is there a similar explanation for the case of $\theta$? Of course, the $\zeta$ function is the $L$-function for the trivial one-dimensional Galois representation, but modular forms should be related to $2$-dimensional Galois representations...

(I've only thought much about the Langlands philosophy quite recently, so I'm probably missing some very basic point!)

EDIT: Thanks to GH from MO for pointing out that the trivial one-dimensional representation corresponds via (a trivial case of) class field theory to the $GL(1)$-automorphic representation defined by the trivial Hecke character.