Properties

Label 2-531-1.1-c7-0-104
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.9·2-s + 195.·4-s + 98.9·5-s + 159.·7-s + 1.21e3·8-s + 1.77e3·10-s + 4.88e3·11-s + 1.16e4·13-s + 2.86e3·14-s − 3.15e3·16-s + 1.04e4·17-s + 1.49e4·19-s + 1.93e4·20-s + 8.78e4·22-s + 88.0·23-s − 6.83e4·25-s + 2.09e5·26-s + 3.11e4·28-s + 1.02e5·29-s − 2.87e5·31-s − 2.12e5·32-s + 1.88e5·34-s + 1.57e4·35-s + 5.66e5·37-s + 2.69e5·38-s + 1.20e5·40-s − 2.96e5·41-s + ⋯
L(s)  = 1  + 1.59·2-s + 1.52·4-s + 0.353·5-s + 0.175·7-s + 0.839·8-s + 0.562·10-s + 1.10·11-s + 1.47·13-s + 0.278·14-s − 0.192·16-s + 0.516·17-s + 0.500·19-s + 0.540·20-s + 1.75·22-s + 0.00150·23-s − 0.874·25-s + 2.34·26-s + 0.268·28-s + 0.781·29-s − 1.73·31-s − 1.14·32-s + 0.821·34-s + 0.0620·35-s + 1.83·37-s + 0.795·38-s + 0.297·40-s − 0.670·41-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: $\chi_{531} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.128689551\)
\(L(\frac12)\) \(\approx\) \(8.128689551\)
\(L(\frac{9}{2})\) 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 \( 1 - 17.9T + 128T^{2} \)
5 \( 1 - 98.9T + 7.81e4T^{2} \)
7 \( 1 - 159.T + 8.23e5T^{2} \)
11 \( 1 - 4.88e3T + 1.94e7T^{2} \)
13 \( 1 - 1.16e4T + 6.27e7T^{2} \)
17 \( 1 - 1.04e4T + 4.10e8T^{2} \)
19 \( 1 - 1.49e4T + 8.93e8T^{2} \)
23 \( 1 - 88.0T + 3.40e9T^{2} \)
29 \( 1 - 1.02e5T + 1.72e10T^{2} \)
31 \( 1 + 2.87e5T + 2.75e10T^{2} \)
37 \( 1 - 5.66e5T + 9.49e10T^{2} \)
41 \( 1 + 2.96e5T + 1.94e11T^{2} \)
43 \( 1 - 7.40e5T + 2.71e11T^{2} \)
47 \( 1 + 9.58e5T + 5.06e11T^{2} \)
53 \( 1 - 7.32e5T + 1.17e12T^{2} \)
61 \( 1 - 2.28e6T + 3.14e12T^{2} \)
67 \( 1 + 4.29e6T + 6.06e12T^{2} \)
71 \( 1 - 4.99e6T + 9.09e12T^{2} \)
73 \( 1 - 3.66e6T + 1.10e13T^{2} \)
79 \( 1 - 3.18e6T + 1.92e13T^{2} \)
83 \( 1 - 4.43e6T + 2.71e13T^{2} \)
89 \( 1 - 1.07e7T + 4.42e13T^{2} \)
97 \( 1 + 4.05e6T + 8.07e13T^{2} \)
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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

−9.738107700895182740562364206484, −8.863823483787141570710266547408, −7.66448918570122578727218252444, −6.45813307897639255383093128406, −5.99270874743642502768405507096, −5.05170859244512996145459493333, −3.93323710185731197166532501292, −3.42044707644478569262811868662, −2.07631149802986628972260898954, −1.04153867641568880216658717569, 1.04153867641568880216658717569, 2.07631149802986628972260898954, 3.42044707644478569262811868662, 3.93323710185731197166532501292, 5.05170859244512996145459493333, 5.99270874743642502768405507096, 6.45813307897639255383093128406, 7.66448918570122578727218252444, 8.863823483787141570710266547408, 9.738107700895182740562364206484

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