Properties

Label 2-531-1.1-c7-0-130
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  + 4.55·2-s − 107.·4-s + 540.·5-s − 1.23e3·7-s − 1.07e3·8-s + 2.46e3·10-s − 3.52e3·11-s − 4.83e3·13-s − 5.64e3·14-s + 8.84e3·16-s − 2.24e3·17-s + 5.31e4·19-s − 5.79e4·20-s − 1.60e4·22-s + 5.63e4·23-s + 2.13e5·25-s − 2.20e4·26-s + 1.32e5·28-s + 1.37e5·29-s − 1.21e5·31-s + 1.77e5·32-s − 1.02e4·34-s − 6.69e5·35-s − 4.76e5·37-s + 2.42e5·38-s − 5.79e5·40-s − 1.40e5·41-s + ⋯
L(s)  = 1  + 0.402·2-s − 0.837·4-s + 1.93·5-s − 1.36·7-s − 0.740·8-s + 0.778·10-s − 0.797·11-s − 0.610·13-s − 0.549·14-s + 0.539·16-s − 0.110·17-s + 1.77·19-s − 1.61·20-s − 0.321·22-s + 0.966·23-s + 2.73·25-s − 0.245·26-s + 1.14·28-s + 1.04·29-s − 0.732·31-s + 0.957·32-s − 0.0446·34-s − 2.63·35-s − 1.54·37-s + 0.715·38-s − 1.43·40-s − 0.318·41-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 \( 1 - 4.55T + 128T^{2} \)
5 \( 1 - 540.T + 7.81e4T^{2} \)
7 \( 1 + 1.23e3T + 8.23e5T^{2} \)
11 \( 1 + 3.52e3T + 1.94e7T^{2} \)
13 \( 1 + 4.83e3T + 6.27e7T^{2} \)
17 \( 1 + 2.24e3T + 4.10e8T^{2} \)
19 \( 1 - 5.31e4T + 8.93e8T^{2} \)
23 \( 1 - 5.63e4T + 3.40e9T^{2} \)
29 \( 1 - 1.37e5T + 1.72e10T^{2} \)
31 \( 1 + 1.21e5T + 2.75e10T^{2} \)
37 \( 1 + 4.76e5T + 9.49e10T^{2} \)
41 \( 1 + 1.40e5T + 1.94e11T^{2} \)
43 \( 1 - 6.96e5T + 2.71e11T^{2} \)
47 \( 1 + 7.83e5T + 5.06e11T^{2} \)
53 \( 1 + 1.05e6T + 1.17e12T^{2} \)
61 \( 1 - 3.78e5T + 3.14e12T^{2} \)
67 \( 1 + 1.12e6T + 6.06e12T^{2} \)
71 \( 1 + 3.82e6T + 9.09e12T^{2} \)
73 \( 1 + 1.68e6T + 1.10e13T^{2} \)
79 \( 1 - 5.56e6T + 1.92e13T^{2} \)
83 \( 1 - 1.80e6T + 2.71e13T^{2} \)
89 \( 1 + 8.26e6T + 4.42e13T^{2} \)
97 \( 1 + 9.36e6T + 8.07e13T^{2} \)
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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

−9.549100172532873763658266987644, −8.751597190249738078452875128750, −7.20841468349879057435239413947, −6.27671321886559272664357804160, −5.42996112346139640538553159259, −4.95663741990869793164724105474, −3.23663197268996183100376602014, −2.70315343055361772063440625363, −1.23274980586705725350414018262, 0, 1.23274980586705725350414018262, 2.70315343055361772063440625363, 3.23663197268996183100376602014, 4.95663741990869793164724105474, 5.42996112346139640538553159259, 6.27671321886559272664357804160, 7.20841468349879057435239413947, 8.751597190249738078452875128750, 9.549100172532873763658266987644

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