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collection.yaml
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collection.yaml
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ID: INPUT{id.yaml}
Title: >
Zeros of Dirichlet L-series
Definition: >
The list contains the first few imaginary parts of zeros of
the first few Dirichlet L-functions $L(\chi,s)$ on the critical line.
Parameters:
q:
title: conductor of $\chi$
type: Z
constraints: $q \geq 1$
n:
title: HREF{#CL}[Conrey index] of $\chi$
type: Z
constraints: $1 \leq n \leq \max(q-1,1)$
comments: >
The pair $(q,n)$ uniquely determines the Dirichlet character $\chi$.
k:
title: Index of the zero
type: Z
constraints: $k \neq 0$
comments: >
The positive imaginary parts are labelled $k = 1, 2, 3, \ldots$,
the negative ones are labelled $k = -1, -2, -3, \ldots$,
and the index $k = 0$ is skipped.
Comments:
comment-Dirichlet-series-definition: >
Let $\chi$ be a Dirichlet character.
The associated L-function $\zeta(\chi,s)$ is the
meromorphic continuation of the Dirichlet series
$\sum_{n=0}^\infty \chi(n) n^{-s}, \Re(s) > 0$.
comment-non-trivial-zeros: >
Its so-called trivial zeros lie on the negative $x$-axis, which are
therefore not listed.
All known non-trivial zeros lie on the line $\Re(s) = 1/2$.
If $\chi$ is real, their imaginary parts are symmetric about 0,
in which case we only list the positive ones.
Formulas:
Programs:
References:
Pla11:
bib: >
David J. Platt, "Computing degree 1 L-functions rigorously",
PhD thesis, University of Bristol, 2011.
Links:
Wiki:
title: "Wikipedia: Dirichlet L-function"
url: https://en.wikipedia.org/wiki/Dirichlet_L-function
LMFDB:
title: "LMFDB: Degree-1 L-functions"
url: https://www.lmfdb.org/L/degree1/
CL:
title: "LMFDB knowl: Conrey label"
url: https://www.lmfdb.org/knowledge/show/character.dirichlet.conrey
Similar tables:
Keywords:
- GRH
Tags:
- zero
- L-function
Data properties:
type: R
complete: no
sources:
- CITE{Pla11}
- CITE{LMFDB}
Display properties:
layout: nested lists
group parameters:
- [q, n]
- [k]
Numbers:
1,1:
equals: HREF{Zeros_of_the_Riemann_zeta_function}
comment: $\chi = 1$
3,2:
both signs: yes
numbers: INPUT{numbers_3_2.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/3/2
4,3:
both signs: yes
numbers: INPUT{numbers_4_3.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/4/3
5,2:
both signs: no
numbers: INPUT{numbers_5_2.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/5/2
5,3:
both signs: no
numbers: INPUT{numbers_5_3.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/5/3
5,4:
both signs: yes
numbers: INPUT{numbers_5_4.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/5/4
7,2:
both signs: no
numbers: INPUT{numbers_7_2.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/7/2
7,3:
both signs: no
numbers: INPUT{numbers_7_3.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/7/3
7,4:
both signs: no
numbers: INPUT{numbers_7_4.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/7/4
7,5:
both signs: no
numbers: INPUT{numbers_7_5.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/7/5
7,6:
both signs: yes
numbers: INPUT{numbers_7_6.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/7/6
8,3:
both signs: yes
numbers: INPUT{numbers_8_3.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/8/5
8,5:
both signs: yes
numbers: INPUT{numbers_8_5.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/8/5
9,2:
both signs: no
numbers: INPUT{numbers_9_2.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/9/2
9,4:
both signs: no
numbers: INPUT{numbers_9_4.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/9/4
9,5:
both signs: no
numbers: INPUT{numbers_9_5.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/9/5
9,7:
both signs: no
numbers: INPUT{numbers_9_7.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/9/7
11,2:
both signs: no
numbers: INPUT{numbers_11_2.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/2
11,3:
both signs: no
numbers: INPUT{numbers_11_3.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/3
11,4:
both signs: no
numbers: INPUT{numbers_11_4.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/4
11,5:
both signs: no
numbers: INPUT{numbers_11_5.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/5
11,6:
both signs: no
numbers: INPUT{numbers_11_6.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/6
11,7:
both signs: no
numbers: INPUT{numbers_11_7.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/7
11,8:
both signs: no
numbers: INPUT{numbers_11_8.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/8
11,9:
both signs: no
numbers: INPUT{numbers_11_9.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/9
11,10:
both signs: yes
numbers: INPUT{numbers_11_10.yaml}
url: https://www.lmfdb.org/Character/Dirichlet/11/10