Artin L-functions

E899965

L-function complex analytic function object of algebraic number theory

Artin L-functions are complex analytic functions attached to Galois representations that generalize Dirichlet L-functions and play a central role in number theory and the study of arithmetic properties of fields.

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Observed surface forms (1)

Surface form	Occurrences
Artin L-function	1

Statements (48)

Predicate	Object
instanceOf	L-function ⓘ complex analytic function ⓘ object of algebraic number theory ⓘ
are	meromorphic functions of a complex variable ⓘ
areAttachedTo	Galois representations ⓘ finite-dimensional complex representations of Galois groups ⓘ representations of the absolute Galois group of a number field ⓘ
associatedWith	Artin representation NERFINISHED ⓘ finite Galois extension of number fields ⓘ
centralRoleIn	Langlands program NERFINISHED ⓘ arithmetic of number fields ⓘ class field theory NERFINISHED ⓘ
conjecturallyEqualTo	automorphic L-functions attached to corresponding automorphic representations ⓘ
conjecturallySatisfy	analytic continuation to entire function except possible pole at s = 1 ⓘ functional equation relating s and 1 - s ⓘ
conjectureAbout	Artin conjecture on nontrivial zeros NERFINISHED ⓘ Artin holomorphy conjecture NERFINISHED ⓘ
definedOver	number fields ⓘ
definedUsing	Artin conductor NERFINISHED ⓘ Euler product ⓘ character of a Galois representation ⓘ local factors at primes ⓘ
domain	complex plane ⓘ
generalize	Dirichlet L-functions NERFINISHED ⓘ
haveParameter	complex variable s ⓘ
haveProperty	compatibility with induction of representations ⓘ multiplicativity with respect to direct sums of representations ⓘ
introducedBy	Emil Artin NERFINISHED ⓘ
localFactorsDefinedBy	Frobenius elements at unramified primes ⓘ
localFactorsModifiedBy	inertia groups at ramified primes ⓘ
namedAfter	Emil Artin NERFINISHED ⓘ
relatedTo	Artin reciprocity law NERFINISHED ⓘ Langlands L-functions NERFINISHED ⓘ automorphic L-functions ⓘ
satisfy	Artin formalism NERFINISHED ⓘ Euler product expansion for Re(s) > 1 ⓘ growth conditions in vertical strips under standard conjectures ⓘ
specialCase	Dedekind zeta function NERFINISHED ⓘ Dirichlet L-function NERFINISHED ⓘ
specialValuesConjecturallyRelatedTo	Stark conjectures NERFINISHED ⓘ algebraic K-theory ⓘ motivic cohomology ⓘ
studyField	number theory ⓘ
usedToStudy	Chebotarev density theorem NERFINISHED ⓘ Galois module structure ⓘ class numbers of number fields ⓘ splitting of primes in extensions ⓘ
yearIntroduced	1920s ⓘ

Referenced by (4)

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

Dirichlet L-functions → relatedTo → Artin L-functions ⓘ

Dedekind zeta functions → relatedTo → Artin L-functions ⓘ

subject surface form: Dedekind zeta function

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

L-functions → hasSpecialCase → Artin L-functions ⓘ

subject surface form: L-function

this entity surface form: Artin L-function