# 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 $0$

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## 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