Properties

Label 2-531-1.1-c7-0-41
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  − 18.7·2-s + 222.·4-s − 443.·5-s − 1.69e3·7-s − 1.76e3·8-s + 8.30e3·10-s + 2.76e3·11-s − 1.09e4·13-s + 3.17e4·14-s + 4.61e3·16-s + 1.71e4·17-s − 4.12e4·19-s − 9.87e4·20-s − 5.17e4·22-s − 8.25e4·23-s + 1.18e5·25-s + 2.05e5·26-s − 3.77e5·28-s − 1.03e5·29-s − 1.62e5·31-s + 1.39e5·32-s − 3.21e5·34-s + 7.52e5·35-s + 4.46e5·37-s + 7.72e5·38-s + 7.84e5·40-s − 3.71e5·41-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.73·4-s − 1.58·5-s − 1.86·7-s − 1.22·8-s + 2.62·10-s + 0.626·11-s − 1.38·13-s + 3.09·14-s + 0.281·16-s + 0.847·17-s − 1.38·19-s − 2.75·20-s − 1.03·22-s − 1.41·23-s + 1.52·25-s + 2.28·26-s − 3.24·28-s − 0.784·29-s − 0.977·31-s + 0.754·32-s − 1.40·34-s + 2.96·35-s + 1.45·37-s + 2.28·38-s + 1.93·40-s − 0.841·41-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 \( 1 + 18.7T + 128T^{2} \)
5 \( 1 + 443.T + 7.81e4T^{2} \)
7 \( 1 + 1.69e3T + 8.23e5T^{2} \)
11 \( 1 - 2.76e3T + 1.94e7T^{2} \)
13 \( 1 + 1.09e4T + 6.27e7T^{2} \)
17 \( 1 - 1.71e4T + 4.10e8T^{2} \)
19 \( 1 + 4.12e4T + 8.93e8T^{2} \)
23 \( 1 + 8.25e4T + 3.40e9T^{2} \)
29 \( 1 + 1.03e5T + 1.72e10T^{2} \)
31 \( 1 + 1.62e5T + 2.75e10T^{2} \)
37 \( 1 - 4.46e5T + 9.49e10T^{2} \)
41 \( 1 + 3.71e5T + 1.94e11T^{2} \)
43 \( 1 + 7.90e5T + 2.71e11T^{2} \)
47 \( 1 - 1.11e6T + 5.06e11T^{2} \)
53 \( 1 - 7.71e5T + 1.17e12T^{2} \)
61 \( 1 + 2.11e5T + 3.14e12T^{2} \)
67 \( 1 - 1.48e6T + 6.06e12T^{2} \)
71 \( 1 + 3.40e6T + 9.09e12T^{2} \)
73 \( 1 + 3.93e6T + 1.10e13T^{2} \)
79 \( 1 - 4.55e6T + 1.92e13T^{2} \)
83 \( 1 - 4.51e6T + 2.71e13T^{2} \)
89 \( 1 - 4.21e6T + 4.42e13T^{2} \)
97 \( 1 + 1.59e6T + 8.07e13T^{2} \)
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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

−9.317023405660320512407622534281, −8.429582145196564365307287593334, −7.53504888254196675978847342968, −7.02731938004571121739008792524, −6.09193938800666538743724587039, −4.21130653437871494744377129271, −3.35816252558520548046078484019, −2.18069913315920447863644747827, −0.53674707995036547970847649452, 0, 0.53674707995036547970847649452, 2.18069913315920447863644747827, 3.35816252558520548046078484019, 4.21130653437871494744377129271, 6.09193938800666538743724587039, 7.02731938004571121739008792524, 7.53504888254196675978847342968, 8.429582145196564365307287593334, 9.317023405660320512407622534281

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