I'm curious about how the L-functions of elliptic curves behave as the elliptic curves vary in families. In other words, if we regard the L-function $L(E_{/K},s)$ of an elliptic curve $E$ over a number field $K$ as a function on the moduli space of elliptic curves over $K$, $\mathcal{M}_{E/K}$, in the first variable, what can be said about its properties, both for fixed values of the second variable $s$, and as a two-variable function on $\mathcal{M}_{E/K} \times \mathbb{C}$?

The question has a natural extension to other moduli spaces of arithmetic varieties.

References to where it is discussed in detail would be appreciated.