Artin L-functions
E899965
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Artin L-functions canonical | 7 |
| Artin L-function | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991153 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Artin L-functions Context triple: [Dirichlet L-functions, relatedTo, Artin L-functions]
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A.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
B.
L-functions
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.
-
C.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
E.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Artin L-functions Target entity description: 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.
-
A.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
B.
L-functions
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.
-
C.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
E.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
- F. None of above. chosen
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
ⓘ
linked to:
Galois representations
finite Galois extension of number fields ⓘ |
| centralRoleIn |
Langlands program
ⓘ
arithmetic of number fields ⓘ class field theory ⓘ
linked to:
global class field theory
|
| 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
ⓘ
linked to:
Artin’s conjecture on L-functions
Artin holomorphy conjecture ⓘ
linked to:
Artin’s conjecture on L-functions
|
| definedOver | number fields ⓘ |
| definedUsing |
Artin conductor
ⓘ
Euler product ⓘ character of a Galois representation ⓘ local factors at primes ⓘ |
| domain | complex plane ⓘ |
| generalize | Dirichlet L-functions ⓘ |
| haveParameter | complex variable s ⓘ |
| haveProperty |
compatibility with induction of representations
ⓘ
multiplicativity with respect to direct sums of representations ⓘ |
| introducedBy | Emil Artin ⓘ |
| localFactorsDefinedBy | Frobenius elements at unramified primes ⓘ |
| localFactorsModifiedBy | inertia groups at ramified primes ⓘ |
| namedAfter | Emil Artin ⓘ |
| relatedTo |
Artin reciprocity law
ⓘ
Langlands L-functions ⓘ
linked to:
L-functions
automorphic L-functions ⓘ |
| satisfy |
Artin formalism
ⓘ
Euler product expansion for Re(s) > 1 ⓘ growth conditions in vertical strips under standard conjectures ⓘ |
| specialCase |
Dedekind zeta function
ⓘ
linked to:
Dedekind zeta functions
Dirichlet L-function ⓘ
linked to:
Dirichlet L-functions
|
| specialValuesConjecturallyRelatedTo |
Stark conjectures
ⓘ
algebraic K-theory ⓘ motivic cohomology ⓘ |
| studyField | number theory ⓘ |
| usedToStudy |
Chebotarev density theorem
ⓘ
Galois module structure ⓘ class numbers of number fields ⓘ splitting of primes in extensions ⓘ |
| yearIntroduced | 1920s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Artin L-functions Description of subject: 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.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Dedekind zeta functions
subject linked to:
L-functions
linked to: Artin L-functions
linked to: Artin L-functions