# Properties

 Label 2-531-59.58-c4-0-65 Degree $2$ Conductor $531$ Sign $0.960 + 0.277i$ 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 + 2.15i·2-s + 11.3·4-s − 30.9·5-s + 73.3·7-s + 58.9i·8-s − 66.6i·10-s − 203. i·11-s + 115. i·13-s + 158. i·14-s + 54.5·16-s + 149.·17-s − 97.5·19-s − 351.·20-s + 438.·22-s − 785. i·23-s + ⋯
 L(s)  = 1 + 0.538i·2-s + 0.709·4-s − 1.23·5-s + 1.49·7-s + 0.921i·8-s − 0.666i·10-s − 1.68i·11-s + 0.681i·13-s + 0.806i·14-s + 0.213·16-s + 0.517·17-s − 0.270·19-s − 0.878·20-s + 0.906·22-s − 1.48i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $0.960 + 0.277i$ 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: $$(2,\ 531,\ (\ :2),\ 0.960 + 0.277i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$2.295033880$$ $$L(\frac12)$$ $$\approx$$ $$2.295033880$$ $$L(3)$$ 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 + (3.34e3 + 964. i)T$$
good2 $$1 - 2.15iT - 16T^{2}$$
5 $$1 + 30.9T + 625T^{2}$$
7 $$1 - 73.3T + 2.40e3T^{2}$$
11 $$1 + 203. iT - 1.46e4T^{2}$$
13 $$1 - 115. iT - 2.85e4T^{2}$$
17 $$1 - 149.T + 8.35e4T^{2}$$
19 $$1 + 97.5T + 1.30e5T^{2}$$
23 $$1 + 785. iT - 2.79e5T^{2}$$
29 $$1 + 279.T + 7.07e5T^{2}$$
31 $$1 + 520. iT - 9.23e5T^{2}$$
37 $$1 + 737. iT - 1.87e6T^{2}$$
41 $$1 + 540.T + 2.82e6T^{2}$$
43 $$1 + 1.17e3iT - 3.41e6T^{2}$$
47 $$1 - 1.50e3iT - 4.87e6T^{2}$$
53 $$1 - 3.63e3T + 7.89e6T^{2}$$
61 $$1 + 5.34e3iT - 1.38e7T^{2}$$
67 $$1 + 1.69e3iT - 2.01e7T^{2}$$
71 $$1 - 9.06e3T + 2.54e7T^{2}$$
73 $$1 - 876. iT - 2.83e7T^{2}$$
79 $$1 - 7.89e3T + 3.89e7T^{2}$$
83 $$1 - 3.53e3iT - 4.74e7T^{2}$$
89 $$1 + 1.14e4iT - 6.27e7T^{2}$$
97 $$1 + 1.51e4iT - 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

−10.65149886439106021082418792626, −8.816898037918575860665467876897, −8.156687393717238292008210413151, −7.71811719758424524079715150158, −6.63769523946134443888880509401, −5.63752182008282626456701694398, −4.57934265155524409952720563847, −3.48522627305143533325073655237, −2.09762622900116108416973577820, −0.63984056594571631474794570034, 1.16139616474366652450633116961, 2.10462444601589398919402014083, 3.48013571930391607540563206154, 4.42383930254365471146793722652, 5.40323946511880730757742189472, 7.03487796935482620504076913517, 7.60556479082490234606526769999, 8.197274039085538593410218379357, 9.632153722560381914986846224720, 10.53405884096909846596336248247