Properties

Label 2-531-1.1-c7-0-131
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  − 7.02·2-s − 78.6·4-s + 266.·5-s + 665.·7-s + 1.45e3·8-s − 1.87e3·10-s − 2.99e3·11-s + 1.11e4·13-s − 4.67e3·14-s − 120.·16-s − 1.73e4·17-s − 7.56e3·19-s − 2.09e4·20-s + 2.10e4·22-s + 8.48e4·23-s − 6.95e3·25-s − 7.80e4·26-s − 5.23e4·28-s − 5.97e4·29-s − 5.62e4·31-s − 1.84e5·32-s + 1.21e5·34-s + 1.77e5·35-s − 3.75e5·37-s + 5.31e4·38-s + 3.87e5·40-s − 5.60e5·41-s + ⋯
L(s)  = 1  − 0.620·2-s − 0.614·4-s + 0.954·5-s + 0.733·7-s + 1.00·8-s − 0.592·10-s − 0.678·11-s + 1.40·13-s − 0.455·14-s − 0.00733·16-s − 0.855·17-s − 0.253·19-s − 0.586·20-s + 0.421·22-s + 1.45·23-s − 0.0890·25-s − 0.870·26-s − 0.450·28-s − 0.454·29-s − 0.338·31-s − 0.997·32-s + 0.530·34-s + 0.700·35-s − 1.21·37-s + 0.157·38-s + 0.956·40-s − 1.26·41-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 \( 1 + 7.02T + 128T^{2} \)
5 \( 1 - 266.T + 7.81e4T^{2} \)
7 \( 1 - 665.T + 8.23e5T^{2} \)
11 \( 1 + 2.99e3T + 1.94e7T^{2} \)
13 \( 1 - 1.11e4T + 6.27e7T^{2} \)
17 \( 1 + 1.73e4T + 4.10e8T^{2} \)
19 \( 1 + 7.56e3T + 8.93e8T^{2} \)
23 \( 1 - 8.48e4T + 3.40e9T^{2} \)
29 \( 1 + 5.97e4T + 1.72e10T^{2} \)
31 \( 1 + 5.62e4T + 2.75e10T^{2} \)
37 \( 1 + 3.75e5T + 9.49e10T^{2} \)
41 \( 1 + 5.60e5T + 1.94e11T^{2} \)
43 \( 1 + 5.99e5T + 2.71e11T^{2} \)
47 \( 1 - 5.08e5T + 5.06e11T^{2} \)
53 \( 1 + 9.51e4T + 1.17e12T^{2} \)
61 \( 1 - 1.08e6T + 3.14e12T^{2} \)
67 \( 1 + 7.31e5T + 6.06e12T^{2} \)
71 \( 1 + 4.30e6T + 9.09e12T^{2} \)
73 \( 1 + 8.04e5T + 1.10e13T^{2} \)
79 \( 1 + 3.53e6T + 1.92e13T^{2} \)
83 \( 1 - 8.87e6T + 2.71e13T^{2} \)
89 \( 1 - 5.83e6T + 4.42e13T^{2} \)
97 \( 1 - 1.47e7T + 8.07e13T^{2} \)
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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

−8.974524062551969222155738155018, −8.717516449489611282296615315251, −7.69641009546881069485839533271, −6.57892450101585065582480243609, −5.44032190281632296829186463238, −4.76647308913372252908196180120, −3.50883841968051350769282064887, −1.98169019823110748198408245775, −1.26922795471849609815744761808, 0, 1.26922795471849609815744761808, 1.98169019823110748198408245775, 3.50883841968051350769282064887, 4.76647308913372252908196180120, 5.44032190281632296829186463238, 6.57892450101585065582480243609, 7.69641009546881069485839533271, 8.717516449489611282296615315251, 8.974524062551969222155738155018

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