Dirichlet L-functions
E259755
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Dirichlet L-functions canonical | 9 |
| Dirichlet L-function | 3 |
| Dirichlet L-function Euler products | 1 |
| Dirichlet L-function modulo q | 1 |
| Dirichlet L-functions modulo q | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364373 — 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: Dirichlet L-functions Context triple: [Riemann hypothesis, relatedTo, Dirichlet L-functions]
-
A.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- 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: Dirichlet L-functions Target entity description: 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.
-
A.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
E.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
L-function
ⓘ
complex analytic function ⓘ object of analytic number theory ⓘ |
| basedOn | Dirichlet characters ⓘ |
| centralIn |
Dirichlet's theorem on arithmetic progressions
ⓘ
surface form:
Dirichlet’s theorem on arithmetic progressions
|
| conjecture |
generalized Riemann hypothesis
ⓘ
surface form:
Generalized Riemann Hypothesis for Dirichlet L-functions
|
| convergesAbsolutelyOn | half-plane Re(s)>1 ⓘ |
| dependsOn |
modulus q of Dirichlet character
ⓘ
primitive or imprimitive nature of character χ ⓘ |
| domain | complex plane ⓘ |
| encodes | arithmetic information about residue classes modulo q ⓘ |
| extendsTo | meromorphic function on C ⓘ |
| generalizes | Riemann zeta function ⓘ |
| GRHStatement | all nontrivial zeros lie on line Re(s)=1/2 ⓘ |
| hasCoefficient | Dirichlet character values χ(n) ⓘ |
| hasComponent | completed L-function Λ(s,χ) ⓘ |
| hasEulerProduct | L(s,χ)=∏_{p}(1−χ(p)p^{-s})^{-1} for Re(s)>1 ⓘ |
| hasGeneralization | L-functions attached to Grössencharacters ⓘ |
| hasSeriesDefinition | L(s,χ)=∑_{n=1}^{∞} χ(n)n^{-s} for Re(s)>1 ⓘ |
| hasSpecialCase |
Dirichlet beta function as L(s,χ) for nontrivial character mod 4
ⓘ
Riemann zeta function ⓘ
surface form:
Riemann zeta function ζ(s)=L(s,χ₀) for trivial character χ₀ modulo 1
|
| hasSymmetry | functional equation symmetric about Re(s)=1/2 ⓘ |
| hasTool | explicit formula relating zeros and primes in arithmetic progressions ⓘ |
| hasZeroType |
nontrivial zeros
ⓘ
trivial zeros ⓘ |
| implies | infinitely many primes in each admissible arithmetic progression ⓘ |
| involves |
Gamma factors in functional equation
ⓘ
conductor of Dirichlet character ⓘ parity of Dirichlet character ⓘ |
| multiplicativityProperty | Dirichlet characters are completely multiplicative ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet ⓘ |
| nontrivialZerosRegion | critical strip 0<Re(s)<1 ⓘ |
| nonvanishingProperty | L(1,χ)≠0 for nontrivial Dirichlet characters χ ⓘ |
| poleProperty | L(s,χ) is entire for nonprincipal characters ⓘ |
| relatedTo |
Artin L-functions
ⓘ
L-functions ⓘ
surface form:
Hecke L-functions
automorphic L-functions ⓘ |
| satisfies |
analytic continuation except possible simple pole at s=1 for principal characters
ⓘ
functional equation relating L(s,χ) and L(1−s,χ̄) ⓘ |
| studiedIn | analytic number theory ⓘ |
| trivialZerosLocation | negative integers depending on parity of χ ⓘ |
| usedIn |
Chebotarev density theorem
ⓘ
analytic class number formula for imaginary quadratic fields ⓘ class number formulas for quadratic fields ⓘ distribution of primes in residue classes ⓘ study of primes in arithmetic progressions ⓘ |
| usedToDefine | Dirichlet density of sets of primes ⓘ |
| variable | complex variable s ⓘ |
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: Dirichlet L-functions Description of subject: 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.
Referenced by (15)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dirichlet L-function
this entity surface form:
Dirichlet L-function modulo q
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function
→
relatedConcept
→
Dirichlet L-functions
ⓘ
this entity surface form:
Dirichlet L-functions modulo q
this entity surface form:
Dirichlet L-function
this entity surface form:
Dirichlet L-function Euler products
subject surface form:
L-function
this entity surface form:
Dirichlet L-function