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Properties

Label 1-4817-4817.4816-r0-0-0
Degree $1$
Conductor $4817$
Sign $1$
Analytic cond. $22.3700$
Root an. cond. $22.3700$
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 − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4817\)
Sign: $1$
Analytic conductor: \(22.3700\)
Root analytic conductor: \(22.3700\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4817} (4816, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4817,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547843324\)
\(L(\frac12)\) \(\approx\) \(1.547843324\)
\(L(1)\) \(\approx\) \(1.146105291\)
\(L(1)\) \(\approx\) \(1.146105291\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4817 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
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

−18.14626640431928711381020949124, −17.26862950927652423541686546848, −16.68795233334404598368823727183, −15.98574164744329477249104624234, −15.39439676841620795783177555121, −14.89539084367839796386412550694, −14.25484448712264446134671143575, −13.1091338385067627437984736137, −12.74704114146519235975035242009, −12.05829817630584047482657545279, −11.446250069473808073212098337029, −10.84141496873603559571960050181, −10.59633527228214458004800668730, −9.3988490763298547070585605099, −8.12837912655454572832263409188, −7.63161914521725296208437614257, −7.12762899949337964172557317886, −6.19302828165717232142448992151, −5.48828664011563060480901604073, −4.714243830699243980626602546427, −4.427297084943139720649932817160, −3.67926000216490965141997027539, −2.37367907888011677211728158270, −1.94017370946814039105401710038, −0.55351926699122008749793838784, 0.55351926699122008749793838784, 1.94017370946814039105401710038, 2.37367907888011677211728158270, 3.67926000216490965141997027539, 4.427297084943139720649932817160, 4.714243830699243980626602546427, 5.48828664011563060480901604073, 6.19302828165717232142448992151, 7.12762899949337964172557317886, 7.63161914521725296208437614257, 8.12837912655454572832263409188, 9.3988490763298547070585605099, 10.59633527228214458004800668730, 10.84141496873603559571960050181, 11.446250069473808073212098337029, 12.05829817630584047482657545279, 12.74704114146519235975035242009, 13.1091338385067627437984736137, 14.25484448712264446134671143575, 14.89539084367839796386412550694, 15.39439676841620795783177555121, 15.98574164744329477249104624234, 16.68795233334404598368823727183, 17.26862950927652423541686546848, 18.14626640431928711381020949124

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