# Properties

 Label 2-531-59.58-c4-0-34 Degree $2$ Conductor $531$ Sign $0.0950 + 0.995i$ Analytic cond. $54.8894$ Root an. cond. $7.40874$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 4.96i·2-s − 8.66·4-s − 41.3·5-s + 1.08·7-s − 36.4i·8-s + 205. i·10-s + 142. i·11-s + 35.2i·13-s − 5.39i·14-s − 319.·16-s − 527.·17-s + 76.0·19-s + 358.·20-s + 708.·22-s − 138. i·23-s + ⋯
 L(s)  = 1 − 1.24i·2-s − 0.541·4-s − 1.65·5-s + 0.0221·7-s − 0.569i·8-s + 2.05i·10-s + 1.17i·11-s + 0.208i·13-s − 0.0275i·14-s − 1.24·16-s − 1.82·17-s + 0.210·19-s + 0.897·20-s + 1.46·22-s − 0.262i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0950 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0950 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $0.0950 + 0.995i$ Analytic conductor: $$54.8894$$ Root analytic conductor: $$7.40874$$ Motivic weight: $$4$$ Rational: no Arithmetic: yes Character: $\chi_{531} (235, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 531,\ (\ :2),\ 0.0950 + 0.995i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.041193801$$ $$L(\frac12)$$ $$\approx$$ $$1.041193801$$ $$L(3)$$ 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 + (330. + 3.46e3i)T$$
good2 $$1 + 4.96iT - 16T^{2}$$
5 $$1 + 41.3T + 625T^{2}$$
7 $$1 - 1.08T + 2.40e3T^{2}$$
11 $$1 - 142. iT - 1.46e4T^{2}$$
13 $$1 - 35.2iT - 2.85e4T^{2}$$
17 $$1 + 527.T + 8.35e4T^{2}$$
19 $$1 - 76.0T + 1.30e5T^{2}$$
23 $$1 + 138. iT - 2.79e5T^{2}$$
29 $$1 - 1.04e3T + 7.07e5T^{2}$$
31 $$1 - 1.36e3iT - 9.23e5T^{2}$$
37 $$1 + 2.07e3iT - 1.87e6T^{2}$$
41 $$1 - 1.60e3T + 2.82e6T^{2}$$
43 $$1 - 746. iT - 3.41e6T^{2}$$
47 $$1 - 613. iT - 4.87e6T^{2}$$
53 $$1 - 3.30e3T + 7.89e6T^{2}$$
61 $$1 + 3.02e3iT - 1.38e7T^{2}$$
67 $$1 - 3.67e3iT - 2.01e7T^{2}$$
71 $$1 + 1.82e3T + 2.54e7T^{2}$$
73 $$1 - 2.69e3iT - 2.83e7T^{2}$$
79 $$1 - 3.08e3T + 3.89e7T^{2}$$
83 $$1 + 7.95e3iT - 4.74e7T^{2}$$
89 $$1 - 5.14e3iT - 6.27e7T^{2}$$
97 $$1 + 1.30e4iT - 8.85e7T^{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

−10.35483619249186139314391453717, −9.269797790394506940056683370696, −8.434755854366738069024821985969, −7.26476652159373395479634233668, −6.71190951830077900521879187534, −4.61940621563684592771913702793, −4.22219343308693966850527432903, −3.08899869873966879742008410400, −1.99136752446078903878055138148, −0.54839391185612415300572686677, 0.52891481922997491268863271095, 2.72942947521523359136221937524, 4.00979867837455175509203473646, 4.86539266936343995170832990379, 6.10860926669132118620474712262, 6.86619578189944252795825368346, 7.78947960916367021629753616128, 8.344895431255666587835778890396, 9.016914632853652672815988012272, 10.67659498485399410394987487360