L-function

An L-function is a Dirichlet series

L (s) = \sum_{n = 1}^{\infty} \frac{a_{n}}{n^{s}}

with three special properties:

Euler product: the coefficients are multiplicative, so the series factors over primes:

L (s) = \prod_{p} (1 - \frac{a_{p}}{p^{s}} + \frac{a_{p^{2}}}{p^{2 s}} - \dots)

Analytic continuation: the series only converges in a half-plane $Re (s) > σ_{c}$ , but extends meromorphically to $C$ .
Functional equation: a symmetry relating $L (s)$ and $L (1 - s)$ through gamma factors.

Examples: the Riemann zeta function ( $a_{n} = 1$ )