Most General Definition of an L-Function

Question

I have seen examples given of $L$-functions, such as Dirichlet $L$-functions and the Riemann Zeta Function, but I have not seen a definition of the most general form of an $L$-function. Basically what I am looking for a set of $L$-functions such that all other L-functions, including automorphic $L$-functions, are elements of that set. A reference to the defining characteristic of such a set would be greatly appreciated.

Answer 1

I believe what you are looking for is the Selberg class of L-functions. This is an axiomatic definition of what we expect from all "reasonable" definitions of L-functions, but it's conjectural that this includes all standard constructions of (complex) L-functions from number theory. On the other hand, Langlands' philosophy conjectures that all conventional L-functions coincide with automorphic L-functions.

Note that there are various other objects called "L-functions" or "zeta functions" outside of what I think of as conventional L/zeta functions from number theory---e.g., people study p-adic L-functions in number theory and various L-functions of graphs in graph theory.