Properties

Label 2-531-1.1-c7-0-70
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  − 14.0·2-s + 69.6·4-s − 153.·5-s − 215.·7-s + 819.·8-s + 2.15e3·10-s − 7.86e3·11-s − 1.42e4·13-s + 3.02e3·14-s − 2.04e4·16-s + 1.44e4·17-s + 5.22e4·19-s − 1.06e4·20-s + 1.10e5·22-s + 2.09e4·23-s − 5.46e4·25-s + 2.00e5·26-s − 1.49e4·28-s + 1.36e5·29-s + 9.32e3·31-s + 1.82e5·32-s − 2.03e5·34-s + 3.29e4·35-s − 5.25e5·37-s − 7.34e5·38-s − 1.25e5·40-s − 1.97e5·41-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.544·4-s − 0.548·5-s − 0.237·7-s + 0.566·8-s + 0.681·10-s − 1.78·11-s − 1.79·13-s + 0.294·14-s − 1.24·16-s + 0.712·17-s + 1.74·19-s − 0.298·20-s + 2.21·22-s + 0.358·23-s − 0.699·25-s + 2.23·26-s − 0.129·28-s + 1.04·29-s + 0.0562·31-s + 0.984·32-s − 0.885·34-s + 0.130·35-s − 1.70·37-s − 2.17·38-s − 0.310·40-s − 0.448·41-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 \( 1 + 14.0T + 128T^{2} \)
5 \( 1 + 153.T + 7.81e4T^{2} \)
7 \( 1 + 215.T + 8.23e5T^{2} \)
11 \( 1 + 7.86e3T + 1.94e7T^{2} \)
13 \( 1 + 1.42e4T + 6.27e7T^{2} \)
17 \( 1 - 1.44e4T + 4.10e8T^{2} \)
19 \( 1 - 5.22e4T + 8.93e8T^{2} \)
23 \( 1 - 2.09e4T + 3.40e9T^{2} \)
29 \( 1 - 1.36e5T + 1.72e10T^{2} \)
31 \( 1 - 9.32e3T + 2.75e10T^{2} \)
37 \( 1 + 5.25e5T + 9.49e10T^{2} \)
41 \( 1 + 1.97e5T + 1.94e11T^{2} \)
43 \( 1 - 4.10e5T + 2.71e11T^{2} \)
47 \( 1 - 9.46e5T + 5.06e11T^{2} \)
53 \( 1 - 1.55e6T + 1.17e12T^{2} \)
61 \( 1 + 6.59e5T + 3.14e12T^{2} \)
67 \( 1 + 2.30e6T + 6.06e12T^{2} \)
71 \( 1 - 1.68e6T + 9.09e12T^{2} \)
73 \( 1 - 2.53e6T + 1.10e13T^{2} \)
79 \( 1 + 1.26e6T + 1.92e13T^{2} \)
83 \( 1 + 2.22e6T + 2.71e13T^{2} \)
89 \( 1 - 1.04e7T + 4.42e13T^{2} \)
97 \( 1 - 5.41e6T + 8.07e13T^{2} \)
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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

−9.377118622718602098466206032977, −8.284306100264883565816520572926, −7.48991908899904279945702596343, −7.27154449096094105321187560668, −5.42333954596901297911734240900, −4.76888234174814502043353974790, −3.19844343463955746568986439793, −2.24290977331084358854346045013, −0.78217434556804683614811557265, 0, 0.78217434556804683614811557265, 2.24290977331084358854346045013, 3.19844343463955746568986439793, 4.76888234174814502043353974790, 5.42333954596901297911734240900, 7.27154449096094105321187560668, 7.48991908899904279945702596343, 8.284306100264883565816520572926, 9.377118622718602098466206032977

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