Properties

Label 2-531-1.1-c7-0-52
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  − 2.11·2-s − 123.·4-s + 537.·5-s + 1.25e3·7-s + 531.·8-s − 1.13e3·10-s − 7.73e3·11-s − 8.86e3·13-s − 2.66e3·14-s + 1.46e4·16-s − 2.23e4·17-s − 2.48e4·19-s − 6.63e4·20-s + 1.63e4·22-s + 2.66e4·23-s + 2.10e5·25-s + 1.87e4·26-s − 1.55e5·28-s − 1.01e4·29-s − 2.65e5·31-s − 9.90e4·32-s + 4.72e4·34-s + 6.76e5·35-s + 1.22e5·37-s + 5.25e4·38-s + 2.85e5·40-s + 7.41e5·41-s + ⋯
L(s)  = 1  − 0.186·2-s − 0.965·4-s + 1.92·5-s + 1.38·7-s + 0.367·8-s − 0.358·10-s − 1.75·11-s − 1.11·13-s − 0.259·14-s + 0.896·16-s − 1.10·17-s − 0.832·19-s − 1.85·20-s + 0.327·22-s + 0.457·23-s + 2.69·25-s + 0.208·26-s − 1.33·28-s − 0.0769·29-s − 1.59·31-s − 0.534·32-s + 0.206·34-s + 2.66·35-s + 0.398·37-s + 0.155·38-s + 0.705·40-s + 1.68·41-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

\(L(4)\) \(\approx\) \(2.170789190\)
\(L(\frac12)\) \(\approx\) \(2.170789190\)
\(L(\frac{9}{2})\) 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 \( 1 + 2.11T + 128T^{2} \)
5 \( 1 - 537.T + 7.81e4T^{2} \)
7 \( 1 - 1.25e3T + 8.23e5T^{2} \)
11 \( 1 + 7.73e3T + 1.94e7T^{2} \)
13 \( 1 + 8.86e3T + 6.27e7T^{2} \)
17 \( 1 + 2.23e4T + 4.10e8T^{2} \)
19 \( 1 + 2.48e4T + 8.93e8T^{2} \)
23 \( 1 - 2.66e4T + 3.40e9T^{2} \)
29 \( 1 + 1.01e4T + 1.72e10T^{2} \)
31 \( 1 + 2.65e5T + 2.75e10T^{2} \)
37 \( 1 - 1.22e5T + 9.49e10T^{2} \)
41 \( 1 - 7.41e5T + 1.94e11T^{2} \)
43 \( 1 + 1.66e5T + 2.71e11T^{2} \)
47 \( 1 - 5.45e5T + 5.06e11T^{2} \)
53 \( 1 - 1.54e6T + 1.17e12T^{2} \)
61 \( 1 - 2.74e6T + 3.14e12T^{2} \)
67 \( 1 + 4.16e4T + 6.06e12T^{2} \)
71 \( 1 - 1.57e6T + 9.09e12T^{2} \)
73 \( 1 - 5.59e6T + 1.10e13T^{2} \)
79 \( 1 + 3.74e6T + 1.92e13T^{2} \)
83 \( 1 - 3.37e6T + 2.71e13T^{2} \)
89 \( 1 - 5.43e6T + 4.42e13T^{2} \)
97 \( 1 + 1.14e7T + 8.07e13T^{2} \)
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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

−9.641673244895876158853730390391, −8.935842084568657015883026060860, −8.110740153967288979304395653672, −7.11568557431518985409874298717, −5.60853590585254494384672038541, −5.20599637666944676862410302704, −4.46033690933065372967858533851, −2.43002195391004696721624622702, −2.01696788746343872146366253082, −0.65054063078390299829798737826, 0.65054063078390299829798737826, 2.01696788746343872146366253082, 2.43002195391004696721624622702, 4.46033690933065372967858533851, 5.20599637666944676862410302704, 5.60853590585254494384672038541, 7.11568557431518985409874298717, 8.110740153967288979304395653672, 8.935842084568657015883026060860, 9.641673244895876158853730390391

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