Editing L-function (section)

== Construction ==

We distinguish at the outset between the '''''L''-series''', an [[infinite set|infinite]] series representation (for example the [[Dirichlet series]] for the [[Riemann zeta function]]), and the '''''L''-function''', the function in the complex plane that is its [[analytic continuation]]. The general constructions start with an ''L''-series, defined first as a [[Dirichlet series]], and then by an expansion as an [[Euler product]] indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the [[complex number]]s. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some [[Pole (complex analysis)|pole]]s).

It is this (conjectural) [[meromorphic]] continuation to the complex plane which is called an ''L''-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the ''L''-function at points where the series representation does not converge. The general term ''L''-function here includes many known types of zeta functions. The [[Selberg class]] is an attempt to capture the core properties of ''L''-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.