Properties

Label 2-531-1.1-c7-0-164
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  + 19.0·2-s + 234.·4-s − 37.6·5-s + 1.15e3·7-s + 2.02e3·8-s − 715.·10-s − 1.61e3·11-s − 7.20e3·13-s + 2.20e4·14-s + 8.48e3·16-s − 2.58e4·17-s − 4.59e4·19-s − 8.80e3·20-s − 3.07e4·22-s + 5.30e4·23-s − 7.67e4·25-s − 1.37e5·26-s + 2.71e5·28-s − 1.64e5·29-s + 6.92e4·31-s − 9.71e4·32-s − 4.91e5·34-s − 4.35e4·35-s − 3.82e5·37-s − 8.75e5·38-s − 7.60e4·40-s + 5.31e5·41-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.82·4-s − 0.134·5-s + 1.27·7-s + 1.39·8-s − 0.226·10-s − 0.366·11-s − 0.909·13-s + 2.14·14-s + 0.517·16-s − 1.27·17-s − 1.53·19-s − 0.246·20-s − 0.616·22-s + 0.909·23-s − 0.981·25-s − 1.52·26-s + 2.33·28-s − 1.25·29-s + 0.417·31-s − 0.524·32-s − 2.14·34-s − 0.171·35-s − 1.24·37-s − 2.58·38-s − 0.187·40-s + 1.20·41-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 \( 1 - 19.0T + 128T^{2} \)
5 \( 1 + 37.6T + 7.81e4T^{2} \)
7 \( 1 - 1.15e3T + 8.23e5T^{2} \)
11 \( 1 + 1.61e3T + 1.94e7T^{2} \)
13 \( 1 + 7.20e3T + 6.27e7T^{2} \)
17 \( 1 + 2.58e4T + 4.10e8T^{2} \)
19 \( 1 + 4.59e4T + 8.93e8T^{2} \)
23 \( 1 - 5.30e4T + 3.40e9T^{2} \)
29 \( 1 + 1.64e5T + 1.72e10T^{2} \)
31 \( 1 - 6.92e4T + 2.75e10T^{2} \)
37 \( 1 + 3.82e5T + 9.49e10T^{2} \)
41 \( 1 - 5.31e5T + 1.94e11T^{2} \)
43 \( 1 - 5.62e5T + 2.71e11T^{2} \)
47 \( 1 + 1.23e6T + 5.06e11T^{2} \)
53 \( 1 + 2.73e5T + 1.17e12T^{2} \)
61 \( 1 - 2.63e6T + 3.14e12T^{2} \)
67 \( 1 + 4.66e6T + 6.06e12T^{2} \)
71 \( 1 - 5.88e6T + 9.09e12T^{2} \)
73 \( 1 + 3.00e6T + 1.10e13T^{2} \)
79 \( 1 - 5.62e6T + 1.92e13T^{2} \)
83 \( 1 - 5.80e6T + 2.71e13T^{2} \)
89 \( 1 - 5.58e6T + 4.42e13T^{2} \)
97 \( 1 - 3.69e6T + 8.07e13T^{2} \)
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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

−9.207659330028578735453428295510, −8.108381979981272780170683158305, −7.20843841413733543852008800960, −6.28663378924193764744130797964, −5.19812710985855383820612626735, −4.64153076990049391718933753762, −3.85959249432967000473662725999, −2.46198419467559134548889482139, −1.88220034610236744903664526886, 0, 1.88220034610236744903664526886, 2.46198419467559134548889482139, 3.85959249432967000473662725999, 4.64153076990049391718933753762, 5.19812710985855383820612626735, 6.28663378924193764744130797964, 7.20843841413733543852008800960, 8.108381979981272780170683158305, 9.207659330028578735453428295510

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