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== Introduction ==

In the realm of [[Number Theory|number theory]], L-functions are a broad class of functions, which are systematically studied due to their deep connections with arithmetic. Named after mathematician [[Edmund Landau|Edmund Landau]], these functions are complex-valued and have a significant role in the study of prime numbers, algebraic number fields, and modular forms.

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== Definition ==

An L-function, in the broadest sense, is a function from the complex numbers to the complex numbers, which satisfies a certain set of conditions. These conditions, which we will describe in detail later, are designed to encapsulate many of the important features of the [[Riemann Zeta Function|Riemann zeta function]] and its generalizations.

== Properties ==

L-functions have several key properties that make them particularly interesting to mathematicians. These include:

* '''Analytic continuation''': L-functions can be extended to functions on the whole complex plane, except possibly for a finite number of poles. This is a property shared by many functions in complex analysis, and it allows us to define the function in regions where it might not initially be defined.
* '''Functional equation''': L-functions satisfy a certain symmetry relation, known as a functional equation, which relates the value of the function at a point s to its value at another point related to s. This is a generalization of the functional equation satisfied by the Riemann zeta function.
* '''Euler product''': L-functions can be expressed as a product over all prime numbers, in a way that generalizes the Euler product formula for the Riemann zeta function. This property reflects the deep connection between L-functions and the distribution of prime numbers.

== Classification ==

L-functions can be classified into several types, based on their source and the nature of their defining data. Some of the most important types include:

* '''Dirichlet L-functions''': These are defined using characters of finite abelian groups, and they generalize the Riemann zeta function in a way that reflects the arithmetic of cyclotomic fields.
* '''Hecke L-functions''': These are associated to [[Hecke Characters|Hecke characters]] of number fields, and they play a key role in class field theory and the study of modular forms.
* '''Automorphic L-functions''': These are associated to automorphic forms, and they are a central object of study in the Langlands program, a far-reaching set of conjectures that link number theory and representation theory.
* '''Artin L-functions''': These are associated to representations of Galois groups, and they are used in the study of algebraic number fields and their Galois groups.

== Applications ==

L-functions have numerous applications in number theory and related fields. Some of the most notable applications include:

* '''Prime number theory''': The Riemann zeta function, which is a special case of an L-function, is deeply connected to the distribution of prime numbers. Generalizations of the Riemann zeta function, such as Dirichlet L-functions, allow us to study primes in arithmetic progressions and other more general settings.
* '''Algebraic number theory''': L-functions associated to number fields, such as Artin L-functions and Hecke L-functions, provide important tools for studying the arithmetic of these fields.
* '''Automorphic forms''': L-functions associated to automorphic forms, such as automorphic L-functions, are central to the study of these forms and their arithmetic properties.

== See Also ==

* [[Riemann Zeta Function]]
* [[Dirichlet L-functions]]
* [[Automorphic Forms]]
* [[Langlands Program]]
* [[Hecke Characters]]

[[Category:Mathematics]]
[[Category:Number Theory]]
[[Category:Complex Analysis]]