TITLE: What is the difference between a zeta function and an L-function?
QUESTION [25 upvotes]: I've been learning about Dedekind zeta functions and some basic L-functions in my introductory algebraic number theory class, and I've been wondering why some functions are called L-functions and others are called zeta functions. I know that the zeta function is a product of L-functions, so it seems like an L-function is somehow a component of a zeta function (at least in the case of Artin L-functions, they correspond to specific representations). Is this the idea behind the distinction between "zeta function" and "L-function"? How do things generalize to other kinds of zeta- and L-functions?

REPLY [11 votes]: I do like the other answers, too, but it seemed silly to append comments to all... :
To my perception, first, I tend to not feel a difference between "zeta function attached to ..." and "L-function attached to..." if only because usage is variable.
Second, in many settings (analytic/automorphic or geometric/motivic or...) a zeta function is an L-function with relatively trivial "further data", whatever that means in context. So, Dedekind zeta functions are Hecke L-functions with trivial data, for example. Analogously for schemes without or with non-trivial sheaf, in the other world. This general rule is certainly not strict... depending on usage.
Third, there are the systematic, partly proven, partly conjectural, miracles that "larger" zetas factor into "smaller" L-functions. Classfield theory and such. This does highlight the ambiguity in "usage", namely, that some "base" is necessary to understand "triviality", etc.
Edit: again significantly contingent on "usage"... If we say that an "L-function" (or "zeta function") "has an analytic continuation (provable by us)", then this would accidentally disallow Hasse-Weil zeta/L-functions of general varieties/schemes/whatever, because in this year we know "few" cases wherein we can prove this, although conjecturally it is mostly-always so (meaning that poles, if any, are finite and describable). Similarly, factorization into Artin L-functions is in one way completely fine (for decades), but, in another, unsatisfactory since we do not know their holomorphy, ... so might decide that they're not yet (in 2012) fully-legitimate "L-functions"? And/or that the "factorization" of Dedekind zetas into such things is not entirely satisfactory (as in a comment). I would not be surprised that such "technicalities" persist in things I know less about. :)