Properties

Label 2-531-1.1-c7-0-123
Degree $2$
Conductor $531$
Sign $-1$
Analytic cond. $165.876$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.44·2-s − 116.·4-s + 382.·5-s − 641.·7-s − 840.·8-s + 1.31e3·10-s − 6.57e3·11-s − 2.08e3·13-s − 2.20e3·14-s + 1.19e4·16-s + 3.72e4·17-s − 1.39e4·19-s − 4.43e4·20-s − 2.26e4·22-s + 8.98e4·23-s + 6.79e4·25-s − 7.17e3·26-s + 7.44e4·28-s − 1.07e5·29-s + 2.18e5·31-s + 1.48e5·32-s + 1.28e5·34-s − 2.45e5·35-s − 2.33e5·37-s − 4.80e4·38-s − 3.20e5·40-s + 5.19e5·41-s + ⋯
L(s)  = 1  + 0.304·2-s − 0.907·4-s + 1.36·5-s − 0.706·7-s − 0.580·8-s + 0.415·10-s − 1.48·11-s − 0.263·13-s − 0.214·14-s + 0.731·16-s + 1.83·17-s − 0.466·19-s − 1.24·20-s − 0.452·22-s + 1.53·23-s + 0.869·25-s − 0.0800·26-s + 0.641·28-s − 0.817·29-s + 1.31·31-s + 0.802·32-s + 0.558·34-s − 0.966·35-s − 0.758·37-s − 0.141·38-s − 0.793·40-s + 1.17·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-1$
Analytic conductor: \(165.876\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 3.44T + 128T^{2} \)
5 \( 1 - 382.T + 7.81e4T^{2} \)
7 \( 1 + 641.T + 8.23e5T^{2} \)
11 \( 1 + 6.57e3T + 1.94e7T^{2} \)
13 \( 1 + 2.08e3T + 6.27e7T^{2} \)
17 \( 1 - 3.72e4T + 4.10e8T^{2} \)
19 \( 1 + 1.39e4T + 8.93e8T^{2} \)
23 \( 1 - 8.98e4T + 3.40e9T^{2} \)
29 \( 1 + 1.07e5T + 1.72e10T^{2} \)
31 \( 1 - 2.18e5T + 2.75e10T^{2} \)
37 \( 1 + 2.33e5T + 9.49e10T^{2} \)
41 \( 1 - 5.19e5T + 1.94e11T^{2} \)
43 \( 1 + 9.07e5T + 2.71e11T^{2} \)
47 \( 1 - 7.59e5T + 5.06e11T^{2} \)
53 \( 1 + 3.80e5T + 1.17e12T^{2} \)
61 \( 1 + 2.37e6T + 3.14e12T^{2} \)
67 \( 1 - 1.91e6T + 6.06e12T^{2} \)
71 \( 1 - 2.83e6T + 9.09e12T^{2} \)
73 \( 1 + 4.60e6T + 1.10e13T^{2} \)
79 \( 1 + 7.37e6T + 1.92e13T^{2} \)
83 \( 1 - 8.05e6T + 2.71e13T^{2} \)
89 \( 1 + 5.97e6T + 4.42e13T^{2} \)
97 \( 1 - 1.56e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.508258675812378070071370605777, −8.507330545772110374319368828730, −7.48382042759560287696063411674, −6.19539517023542490222874671938, −5.45756468604768846671396831562, −4.88581630587227747121463118881, −3.34496791124621879204006749003, −2.62267260962162363091255841240, −1.17698809274804263402409721326, 0, 1.17698809274804263402409721326, 2.62267260962162363091255841240, 3.34496791124621879204006749003, 4.88581630587227747121463118881, 5.45756468604768846671396831562, 6.19539517023542490222874671938, 7.48382042759560287696063411674, 8.507330545772110374319368828730, 9.508258675812378070071370605777

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