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Properties

Label 1-5269-5269.5268-r0-0-0
Degree $1$
Conductor $5269$
Sign $1$
Analytic cond. $24.4691$
Root an. cond. $24.4691$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s − 30-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 21-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + 29-s − 30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5269\)    =    \(11 \cdot 479\)
Sign: $1$
Analytic conductor: \(24.4691\)
Root analytic conductor: \(24.4691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5269} (5268, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 5269,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.441272559\)
\(L(\frac12)\) \(\approx\) \(2.441272559\)
\(L(1)\) \(\approx\) \(1.293706842\)
\(L(1)\) \(\approx\) \(1.293706842\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
479 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.104513504058491589945463552447, −17.41740988635523202629600032213, −16.509165687879550021361613114320, −16.09659107257434025167896431127, −15.52183703351976028232295190859, −14.60587340221568088390007317329, −14.042849158797760669260035014229, −13.32663241237866873250136591390, −12.6870918635868121125386363828, −12.050129809957612558523208578332, −10.78270712423768647491423951304, −10.45193731051054511922597009369, −9.55067747507836675630289705656, −9.30042961812240418998325382091, −8.7428738812363337731481732405, −7.81009322473199440383193618380, −7.22673056482893334051647355617, −6.43325004961848207535210426908, −5.91693667977431392517478358265, −4.950561437578136147825714568468, −3.47322955945978808955248277110, −3.20567914639520495183231905013, −2.44817378707135778604281212413, −1.473222768208079772366279520984, −0.98121429036693043208060697213, 0.98121429036693043208060697213, 1.473222768208079772366279520984, 2.44817378707135778604281212413, 3.20567914639520495183231905013, 3.47322955945978808955248277110, 4.950561437578136147825714568468, 5.91693667977431392517478358265, 6.43325004961848207535210426908, 7.22673056482893334051647355617, 7.81009322473199440383193618380, 8.7428738812363337731481732405, 9.30042961812240418998325382091, 9.55067747507836675630289705656, 10.45193731051054511922597009369, 10.78270712423768647491423951304, 12.050129809957612558523208578332, 12.6870918635868121125386363828, 13.32663241237866873250136591390, 14.042849158797760669260035014229, 14.60587340221568088390007317329, 15.52183703351976028232295190859, 16.09659107257434025167896431127, 16.509165687879550021361613114320, 17.41740988635523202629600032213, 18.104513504058491589945463552447

Graph of the $Z$-function along the critical line