Properties

Label 2-531-1.1-c7-0-1
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 $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 16.2·2-s + 136.·4-s + 30.6·5-s − 1.35e3·7-s − 137.·8-s − 498.·10-s − 1.82e3·11-s − 1.17e3·13-s + 2.20e4·14-s − 1.52e4·16-s − 1.31e4·17-s − 1.85e4·19-s + 4.18e3·20-s + 2.96e4·22-s + 1.45e4·23-s − 7.71e4·25-s + 1.90e4·26-s − 1.85e5·28-s + 1.83e5·29-s − 2.65e5·31-s + 2.65e5·32-s + 2.14e5·34-s − 4.16e4·35-s − 2.60e5·37-s + 3.02e5·38-s − 4.21e3·40-s − 7.19e5·41-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.06·4-s + 0.109·5-s − 1.49·7-s − 0.0948·8-s − 0.157·10-s − 0.413·11-s − 0.148·13-s + 2.14·14-s − 0.929·16-s − 0.650·17-s − 0.621·19-s + 0.116·20-s + 0.594·22-s + 0.249·23-s − 0.987·25-s + 0.212·26-s − 1.59·28-s + 1.39·29-s − 1.59·31-s + 1.43·32-s + 0.935·34-s − 0.164·35-s − 0.846·37-s + 0.893·38-s − 0.0104·40-s − 1.62·41-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: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.006111213413\)
\(L(\frac12)\) \(\approx\) \(0.006111213413\)
\(L(\frac{9}{2})\) 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 \( 1 + 16.2T + 128T^{2} \)
5 \( 1 - 30.6T + 7.81e4T^{2} \)
7 \( 1 + 1.35e3T + 8.23e5T^{2} \)
11 \( 1 + 1.82e3T + 1.94e7T^{2} \)
13 \( 1 + 1.17e3T + 6.27e7T^{2} \)
17 \( 1 + 1.31e4T + 4.10e8T^{2} \)
19 \( 1 + 1.85e4T + 8.93e8T^{2} \)
23 \( 1 - 1.45e4T + 3.40e9T^{2} \)
29 \( 1 - 1.83e5T + 1.72e10T^{2} \)
31 \( 1 + 2.65e5T + 2.75e10T^{2} \)
37 \( 1 + 2.60e5T + 9.49e10T^{2} \)
41 \( 1 + 7.19e5T + 1.94e11T^{2} \)
43 \( 1 + 7.07e5T + 2.71e11T^{2} \)
47 \( 1 + 1.30e6T + 5.06e11T^{2} \)
53 \( 1 - 1.80e6T + 1.17e12T^{2} \)
61 \( 1 - 1.38e6T + 3.14e12T^{2} \)
67 \( 1 - 2.41e6T + 6.06e12T^{2} \)
71 \( 1 + 2.72e6T + 9.09e12T^{2} \)
73 \( 1 - 3.97e6T + 1.10e13T^{2} \)
79 \( 1 + 6.91e6T + 1.92e13T^{2} \)
83 \( 1 + 8.32e6T + 2.71e13T^{2} \)
89 \( 1 - 6.71e6T + 4.42e13T^{2} \)
97 \( 1 + 4.77e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.844267291613145843432957577368, −8.831112580720729037794868277690, −8.236307554774886648463525599060, −6.96041177437166431277129734635, −6.60257507091394880181078447058, −5.23741790567753423793845404598, −3.81262544153021901046796735507, −2.64146954508105301578007370972, −1.59153133506467045143530716566, −0.04150207403620255132075027165, 0.04150207403620255132075027165, 1.59153133506467045143530716566, 2.64146954508105301578007370972, 3.81262544153021901046796735507, 5.23741790567753423793845404598, 6.60257507091394880181078447058, 6.96041177437166431277129734635, 8.236307554774886648463525599060, 8.831112580720729037794868277690, 9.844267291613145843432957577368

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