Properties

Label 2-531-1.1-c7-0-109
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  − 3.09·2-s − 118.·4-s + 156.·5-s + 11.3·7-s + 763.·8-s − 484.·10-s − 4.29e3·11-s + 1.74e3·13-s − 35.1·14-s + 1.27e4·16-s + 4.76e3·17-s + 1.82e4·19-s − 1.85e4·20-s + 1.33e4·22-s − 8.20e4·23-s − 5.36e4·25-s − 5.39e3·26-s − 1.34e3·28-s + 1.57e5·29-s + 3.77e4·31-s − 1.37e5·32-s − 1.47e4·34-s + 1.77e3·35-s + 4.50e5·37-s − 5.66e4·38-s + 1.19e5·40-s + 4.37e5·41-s + ⋯
L(s)  = 1  − 0.273·2-s − 0.925·4-s + 0.559·5-s + 0.0125·7-s + 0.527·8-s − 0.153·10-s − 0.973·11-s + 0.219·13-s − 0.00342·14-s + 0.780·16-s + 0.235·17-s + 0.611·19-s − 0.517·20-s + 0.266·22-s − 1.40·23-s − 0.686·25-s − 0.0601·26-s − 0.0115·28-s + 1.19·29-s + 0.227·31-s − 0.740·32-s − 0.0643·34-s + 0.00700·35-s + 1.46·37-s − 0.167·38-s + 0.294·40-s + 0.991·41-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: \(1\)
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 \( 1 + 3.09T + 128T^{2} \)
5 \( 1 - 156.T + 7.81e4T^{2} \)
7 \( 1 - 11.3T + 8.23e5T^{2} \)
11 \( 1 + 4.29e3T + 1.94e7T^{2} \)
13 \( 1 - 1.74e3T + 6.27e7T^{2} \)
17 \( 1 - 4.76e3T + 4.10e8T^{2} \)
19 \( 1 - 1.82e4T + 8.93e8T^{2} \)
23 \( 1 + 8.20e4T + 3.40e9T^{2} \)
29 \( 1 - 1.57e5T + 1.72e10T^{2} \)
31 \( 1 - 3.77e4T + 2.75e10T^{2} \)
37 \( 1 - 4.50e5T + 9.49e10T^{2} \)
41 \( 1 - 4.37e5T + 1.94e11T^{2} \)
43 \( 1 + 7.38e5T + 2.71e11T^{2} \)
47 \( 1 + 8.69e5T + 5.06e11T^{2} \)
53 \( 1 - 9.91e5T + 1.17e12T^{2} \)
61 \( 1 + 1.30e6T + 3.14e12T^{2} \)
67 \( 1 - 2.85e6T + 6.06e12T^{2} \)
71 \( 1 - 4.56e6T + 9.09e12T^{2} \)
73 \( 1 - 9.37e5T + 1.10e13T^{2} \)
79 \( 1 - 6.61e6T + 1.92e13T^{2} \)
83 \( 1 - 3.65e6T + 2.71e13T^{2} \)
89 \( 1 - 5.71e6T + 4.42e13T^{2} \)
97 \( 1 + 1.38e7T + 8.07e13T^{2} \)
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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

−9.544683620961864393955370245252, −8.164671220004109018647729906251, −7.928849390706178882103925810474, −6.40955302227805949812460219710, −5.48687874388228137046734275219, −4.67539935579422782340379386523, −3.54987510861044648162063384557, −2.29816933464404822591477315531, −1.07454455910264601714886975843, 0, 1.07454455910264601714886975843, 2.29816933464404822591477315531, 3.54987510861044648162063384557, 4.67539935579422782340379386523, 5.48687874388228137046734275219, 6.40955302227805949812460219710, 7.928849390706178882103925810474, 8.164671220004109018647729906251, 9.544683620961864393955370245252

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