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 | 3 |
| Artin L-function | 1 |
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.
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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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Dedekind zeta function
subject surface form:
L-function
this entity surface form:
Artin L-function