Properties

Label 2-531-1.1-c7-0-95
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.01·2-s − 91.8·4-s + 385.·5-s + 847.·7-s − 1.32e3·8-s + 2.32e3·10-s + 5.52e3·11-s + 2.98e3·13-s + 5.09e3·14-s + 3.79e3·16-s + 6.65e3·17-s + 4.74e4·19-s − 3.54e4·20-s + 3.32e4·22-s − 1.69e4·23-s + 7.07e4·25-s + 1.79e4·26-s − 7.78e4·28-s − 2.63e3·29-s + 1.38e5·31-s + 1.92e5·32-s + 4.00e4·34-s + 3.27e5·35-s − 9.12e4·37-s + 2.85e5·38-s − 5.10e5·40-s − 5.49e5·41-s + ⋯
L(s)  = 1  + 0.531·2-s − 0.717·4-s + 1.38·5-s + 0.934·7-s − 0.913·8-s + 0.733·10-s + 1.25·11-s + 0.377·13-s + 0.496·14-s + 0.231·16-s + 0.328·17-s + 1.58·19-s − 0.990·20-s + 0.665·22-s − 0.289·23-s + 0.905·25-s + 0.200·26-s − 0.670·28-s − 0.0200·29-s + 0.836·31-s + 1.03·32-s + 0.174·34-s + 1.28·35-s − 0.296·37-s + 0.843·38-s − 1.26·40-s − 1.24·41-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

\(L(4)\) \(\approx\) \(4.613869580\)
\(L(\frac12)\) \(\approx\) \(4.613869580\)
\(L(\frac{9}{2})\) 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 \( 1 - 6.01T + 128T^{2} \)
5 \( 1 - 385.T + 7.81e4T^{2} \)
7 \( 1 - 847.T + 8.23e5T^{2} \)
11 \( 1 - 5.52e3T + 1.94e7T^{2} \)
13 \( 1 - 2.98e3T + 6.27e7T^{2} \)
17 \( 1 - 6.65e3T + 4.10e8T^{2} \)
19 \( 1 - 4.74e4T + 8.93e8T^{2} \)
23 \( 1 + 1.69e4T + 3.40e9T^{2} \)
29 \( 1 + 2.63e3T + 1.72e10T^{2} \)
31 \( 1 - 1.38e5T + 2.75e10T^{2} \)
37 \( 1 + 9.12e4T + 9.49e10T^{2} \)
41 \( 1 + 5.49e5T + 1.94e11T^{2} \)
43 \( 1 + 2.40e5T + 2.71e11T^{2} \)
47 \( 1 - 1.17e6T + 5.06e11T^{2} \)
53 \( 1 - 1.00e5T + 1.17e12T^{2} \)
61 \( 1 - 7.14e4T + 3.14e12T^{2} \)
67 \( 1 + 9.12e5T + 6.06e12T^{2} \)
71 \( 1 - 2.74e5T + 9.09e12T^{2} \)
73 \( 1 + 5.34e6T + 1.10e13T^{2} \)
79 \( 1 - 1.12e6T + 1.92e13T^{2} \)
83 \( 1 - 4.95e6T + 2.71e13T^{2} \)
89 \( 1 - 8.64e6T + 4.42e13T^{2} \)
97 \( 1 + 2.69e6T + 8.07e13T^{2} \)
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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

−9.581091357983498822896560783600, −9.016685845300106981243433739000, −8.069343349235523069442252716933, −6.72681297719858340332880470623, −5.76419560369578070183892446017, −5.18741532059420825150603962231, −4.17359949770006719696261434222, −3.09331740522763421465591664076, −1.69298751127431239938567328625, −0.961940067735263229870880578500, 0.961940067735263229870880578500, 1.69298751127431239938567328625, 3.09331740522763421465591664076, 4.17359949770006719696261434222, 5.18741532059420825150603962231, 5.76419560369578070183892446017, 6.72681297719858340332880470623, 8.069343349235523069442252716933, 9.016685845300106981243433739000, 9.581091357983498822896560783600

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