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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
  number-header: >
    $k$<sup>th</sup> zero of $L(\chi_q(n,\cdot),s)$
  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

— preview —

Zeros of Dirichlet L-series

edit on github

Numbers

$q$, $n$

$k$

$k$^th zero of $L(\chi_q(n,\cdot),s)$

1, 1:

equals: Zeros_of_the_Riemann_zeta_function
comment: $\chi = 1$

3, 2

INPUT{numbers_3_2.yaml} (not shown in preview)

4, 3

INPUT{numbers_4_3.yaml} (not shown in preview)

5, 2

INPUT{numbers_5_2.yaml} (not shown in preview)

5, 3

INPUT{numbers_5_3.yaml} (not shown in preview)

5, 4

INPUT{numbers_5_4.yaml} (not shown in preview)

7, 2

INPUT{numbers_7_2.yaml} (not shown in preview)

7, 3

INPUT{numbers_7_3.yaml} (not shown in preview)

7, 4

INPUT{numbers_7_4.yaml} (not shown in preview)

7, 5

INPUT{numbers_7_5.yaml} (not shown in preview)

7, 6

INPUT{numbers_7_6.yaml} (not shown in preview)

8, 3

INPUT{numbers_8_3.yaml} (not shown in preview)

8, 5

INPUT{numbers_8_5.yaml} (not shown in preview)

9, 2

INPUT{numbers_9_2.yaml} (not shown in preview)

9, 4

INPUT{numbers_9_4.yaml} (not shown in preview)

9, 5

INPUT{numbers_9_5.yaml} (not shown in preview)

9, 7

INPUT{numbers_9_7.yaml} (not shown in preview)

11, 2

INPUT{numbers_11_2.yaml} (not shown in preview)

11, 3

INPUT{numbers_11_3.yaml} (not shown in preview)

11, 4

INPUT{numbers_11_4.yaml} (not shown in preview)

11, 5

INPUT{numbers_11_5.yaml} (not shown in preview)

11, 6

INPUT{numbers_11_6.yaml} (not shown in preview)

11, 7

INPUT{numbers_11_7.yaml} (not shown in preview)

11, 8

INPUT{numbers_11_8.yaml} (not shown in preview)

11, 9

INPUT{numbers_11_9.yaml} (not shown in preview)

11, 10

INPUT{numbers_11_10.yaml} (not shown in preview)

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

$q$

— conductor of $\chi$ ($q \geq 1$)

$n$

— Conrey index of $\chi$ ($1 \leq n \leq \max(q-1,1)$)

$k$

— Index of the zero ($k \neq 0$)

Comments

(1)

Let $\chi$ be a Dirichlet character. The associated L-function $\zeta(\chi,s)$ is the meromorphic continuation of the Dirichlet series $\sum_{n=0}^\infty \chi(n) n^{-s}, \Re(s) > 0$.

(2)

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.

References

[1]

David J. Platt, "Computing degree 1 L-functions rigorously", PhD thesis, University of Bristol, 2011.

Links

[2]

Wikipedia: Dirichlet L-function

[3]

LMFDB: Degree-1 L-functions

[4]

LMFDB knowl: Conrey label

Data properties

Entries are of type: real number

Table is complete: no

Sources of data: [1], [3]