L-functions

E358024

complex analytic function mathematical object

L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.

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All labels observed (6)

Label	Occurrences
L-functions canonical	5
Hecke L-functions	3
Artin L-functions	1
Artin zeta function	1
Hecke L-function	1
Rankin–Selberg L-functions	1

Statements (49)

Predicate	Object
instanceOf	complex analytic function ⓘ mathematical object ⓘ
associatedWith	algebraic varieties ⓘ automorphic representations ⓘ characters of Galois groups ⓘ number fields ⓘ
conjecturallySatisfies	generalized Riemann hypothesis ⓘ
definedBy	infinite series sum a_n n^{-s} ⓘ
encodes	arithmetic information ⓘ
field	algebraic geometry ⓘ number theory ⓘ
generalizes	Riemann zeta function ⓘ
hasCriticalStrip	0 < Re(s) < 1 ⓘ
hasDomain	complex plane ⓘ
hasInvariant	conductor ⓘ degree ⓘ gamma factor ⓘ root number ⓘ
hasProperty	analytic continuation conjectured or proved beyond region of convergence ⓘ
hasSpecialCase	Artin L-functions ⓘ surface form: Artin L-function Dedekind zeta functions ⓘ surface form: Dedekind zeta function Dirichlet L-functions ⓘ surface form: Dirichlet L-function Hasse–Weil zeta function ⓘ surface form: Hasse–Weil L-function L-functions self-linksurface differs ⓘ surface form: Hecke L-function L-function of a modular form ⓘ L-function of an elliptic curve ⓘ Rankin–Selberg L-function ⓘ Selberg zeta function ⓘ
is	meromorphic function on the complex plane ⓘ
oftenDefinedBy	Dirichlet series ⓘ
oftenHas	Euler product expansion ⓘ
oftenNormalizedTo	satisfy symmetric functional equation ⓘ
playsRoleIn	Birch and Swinnerton-Dyer Conjecture ⓘ surface form: Birch and Swinnerton-Dyer conjecture class number formulas ⓘ modularity theorem ⓘ prime number theorem generalizations ⓘ
relatedTo	Galois representations ⓘ automorphic forms ⓘ motives ⓘ prime numbers ⓘ
satisfies	functional equation relating s and 1-s ⓘ
studiedBy	André Weil ⓘ Bernhard Riemann ⓘ Erich Hecke ⓘ Robert Langlands ⓘ
usedIn	Langlands program ⓘ analytic number theory ⓘ arithmetic geometry ⓘ
zerosConjecturallyLieOn	critical line Re(s)=1/2 ⓘ

Referenced by (12)

Full triples — surface form annotated when it differs from this entity's canonical label.

Karl Rubin → researchArea → L-functions ⓘ

Emil Artin → notableWork → L-functions ⓘ

this entity surface form: Artin L-functions

Emil Artin → notableWork → L-functions ⓘ

this entity surface form: Artin zeta function

Multiplicative Number Theory → usesConcept → L-functions ⓘ

Hasse–Weil zeta function → relatedTo → L-functions ⓘ

Selberg class → contains → L-functions ⓘ

this entity surface form: Hecke L-functions

Dirichlet L-functions → relatedTo → L-functions ⓘ

this entity surface form: Hecke L-functions

Dedekind zeta functions → relatedTo → L-functions ⓘ

subject surface form: Dedekind zeta function

this entity surface form: Hecke L-functions

Representation Theory and Automorphic Functions → topic → L-functions ⓘ

Dirichlet series → isUsedToStudy → L-functions ⓘ

Euler products for automorphic L-functions → relatedTo → L-functions ⓘ

this entity surface form: Rankin–Selberg L-functions

L-functions → hasSpecialCase → L-functions self-linksurface differs ⓘ

subject surface form: L-function

this entity surface form: Hecke L-function