Properties

Label 2-531-1.1-c7-0-60
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  + 12.1·2-s + 19.3·4-s − 236.·5-s + 1.42e3·7-s − 1.31e3·8-s − 2.86e3·10-s + 5.47e3·11-s − 8.45e3·13-s + 1.73e4·14-s − 1.84e4·16-s + 6.08e3·17-s − 1.29e4·19-s − 4.57e3·20-s + 6.64e4·22-s + 5.39e4·23-s − 2.22e4·25-s − 1.02e5·26-s + 2.76e4·28-s + 1.43e4·29-s − 4.84e4·31-s − 5.56e4·32-s + 7.38e4·34-s − 3.37e5·35-s + 8.28e4·37-s − 1.57e5·38-s + 3.11e5·40-s + 7.82e5·41-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.151·4-s − 0.845·5-s + 1.57·7-s − 0.910·8-s − 0.907·10-s + 1.23·11-s − 1.06·13-s + 1.68·14-s − 1.12·16-s + 0.300·17-s − 0.433·19-s − 0.127·20-s + 1.33·22-s + 0.925·23-s − 0.285·25-s − 1.14·26-s + 0.237·28-s + 0.109·29-s − 0.292·31-s − 0.299·32-s + 0.322·34-s − 1.32·35-s + 0.269·37-s − 0.465·38-s + 0.770·40-s + 1.77·41-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

\(L(4)\) \(\approx\) \(3.509316315\)
\(L(\frac12)\) \(\approx\) \(3.509316315\)
\(L(\frac{9}{2})\) 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 \( 1 - 12.1T + 128T^{2} \)
5 \( 1 + 236.T + 7.81e4T^{2} \)
7 \( 1 - 1.42e3T + 8.23e5T^{2} \)
11 \( 1 - 5.47e3T + 1.94e7T^{2} \)
13 \( 1 + 8.45e3T + 6.27e7T^{2} \)
17 \( 1 - 6.08e3T + 4.10e8T^{2} \)
19 \( 1 + 1.29e4T + 8.93e8T^{2} \)
23 \( 1 - 5.39e4T + 3.40e9T^{2} \)
29 \( 1 - 1.43e4T + 1.72e10T^{2} \)
31 \( 1 + 4.84e4T + 2.75e10T^{2} \)
37 \( 1 - 8.28e4T + 9.49e10T^{2} \)
41 \( 1 - 7.82e5T + 1.94e11T^{2} \)
43 \( 1 + 3.69e5T + 2.71e11T^{2} \)
47 \( 1 + 3.68e5T + 5.06e11T^{2} \)
53 \( 1 + 8.36e5T + 1.17e12T^{2} \)
61 \( 1 - 3.73e4T + 3.14e12T^{2} \)
67 \( 1 - 2.64e6T + 6.06e12T^{2} \)
71 \( 1 - 2.03e6T + 9.09e12T^{2} \)
73 \( 1 + 3.64e6T + 1.10e13T^{2} \)
79 \( 1 + 7.63e6T + 1.92e13T^{2} \)
83 \( 1 - 6.69e6T + 2.71e13T^{2} \)
89 \( 1 + 7.17e5T + 4.42e13T^{2} \)
97 \( 1 - 1.06e7T + 8.07e13T^{2} \)
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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

−9.604039936226085402350340211910, −8.711844479232959858762158583553, −7.84829980838393717066508860538, −6.95969598903545628460500144245, −5.75499031178273349502217104875, −4.71518435606599876628924331534, −4.34132858279796193295116843136, −3.30664027907559684805221561077, −1.99180000255376913210499705975, −0.69896490452828524142793584359, 0.69896490452828524142793584359, 1.99180000255376913210499705975, 3.30664027907559684805221561077, 4.34132858279796193295116843136, 4.71518435606599876628924331534, 5.75499031178273349502217104875, 6.95969598903545628460500144245, 7.84829980838393717066508860538, 8.711844479232959858762158583553, 9.604039936226085402350340211910

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