Properties

Label 2-531-1.1-c7-0-108
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  − 11.4·2-s + 3.05·4-s − 248.·5-s + 1.06e3·7-s + 1.43e3·8-s + 2.84e3·10-s + 5.87e3·11-s − 9.75e3·13-s − 1.22e4·14-s − 1.67e4·16-s + 2.32e4·17-s + 2.53e4·19-s − 757.·20-s − 6.72e4·22-s − 7.13e4·23-s − 1.64e4·25-s + 1.11e5·26-s + 3.25e3·28-s − 2.14e5·29-s + 1.53e5·31-s + 8.83e3·32-s − 2.65e5·34-s − 2.64e5·35-s − 1.04e5·37-s − 2.89e5·38-s − 3.55e5·40-s − 2.10e5·41-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.0238·4-s − 0.888·5-s + 1.17·7-s + 0.987·8-s + 0.899·10-s + 1.33·11-s − 1.23·13-s − 1.18·14-s − 1.02·16-s + 1.14·17-s + 0.846·19-s − 0.0211·20-s − 1.34·22-s − 1.22·23-s − 0.210·25-s + 1.24·26-s + 0.0280·28-s − 1.63·29-s + 0.925·31-s + 0.0476·32-s − 1.16·34-s − 1.04·35-s − 0.338·37-s − 0.856·38-s − 0.877·40-s − 0.477·41-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 \( 1 + 11.4T + 128T^{2} \)
5 \( 1 + 248.T + 7.81e4T^{2} \)
7 \( 1 - 1.06e3T + 8.23e5T^{2} \)
11 \( 1 - 5.87e3T + 1.94e7T^{2} \)
13 \( 1 + 9.75e3T + 6.27e7T^{2} \)
17 \( 1 - 2.32e4T + 4.10e8T^{2} \)
19 \( 1 - 2.53e4T + 8.93e8T^{2} \)
23 \( 1 + 7.13e4T + 3.40e9T^{2} \)
29 \( 1 + 2.14e5T + 1.72e10T^{2} \)
31 \( 1 - 1.53e5T + 2.75e10T^{2} \)
37 \( 1 + 1.04e5T + 9.49e10T^{2} \)
41 \( 1 + 2.10e5T + 1.94e11T^{2} \)
43 \( 1 - 5.36e5T + 2.71e11T^{2} \)
47 \( 1 + 7.98e5T + 5.06e11T^{2} \)
53 \( 1 + 5.11e4T + 1.17e12T^{2} \)
61 \( 1 - 1.19e6T + 3.14e12T^{2} \)
67 \( 1 - 2.40e6T + 6.06e12T^{2} \)
71 \( 1 + 3.05e6T + 9.09e12T^{2} \)
73 \( 1 + 4.01e6T + 1.10e13T^{2} \)
79 \( 1 - 8.12e6T + 1.92e13T^{2} \)
83 \( 1 - 9.00e6T + 2.71e13T^{2} \)
89 \( 1 + 6.67e5T + 4.42e13T^{2} \)
97 \( 1 + 8.52e5T + 8.07e13T^{2} \)
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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

−9.368205743642470017548347311896, −8.136432667770211969968872997953, −7.83950954164426140918528333454, −7.03123805039238211652438950684, −5.44891198577196380707743109017, −4.47284717929894815318390112282, −3.66825377978734843882141691540, −1.92053493791234901396899822106, −1.06722605181088207063597161218, 0, 1.06722605181088207063597161218, 1.92053493791234901396899822106, 3.66825377978734843882141691540, 4.47284717929894815318390112282, 5.44891198577196380707743109017, 7.03123805039238211652438950684, 7.83950954164426140918528333454, 8.136432667770211969968872997953, 9.368205743642470017548347311896

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