The term 
-function
generally denotes a complex function represented
in some right half-plane by a Dirichlet
series
 |
whose coefficients encode arithmetic data. Standard arithmetic examples also have an Euler product
 |
where 
ranges over the prime
numbers and each 
is a polynomial. Although there is no single definition
covering every usage, the principal classes of 
-functions admit, or are expected to admit, analytic
continuation and a functional equation.
Such a functional equation determines a critical strip and central
point, and the zeros of the 
-function encode arithmetic information.
The Riemann zeta function is the basic example. Other important families include Artin L-functions, Dirichlet L-series, Hasse-Weil L-functions, and Hecke L-functions.
-Function."