# Properties

 Label 2-531-59.58-c4-0-10 Degree $2$ Conductor $531$ Sign $0.486 + 0.873i$ Analytic cond. $54.8894$ Root an. cond. $7.40874$ Motivic weight $4$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7.65i·2-s − 42.5·4-s + 38.1·5-s − 35.1·7-s − 203. i·8-s + 292. i·10-s + 147. i·11-s − 17.0i·13-s − 268. i·14-s + 875.·16-s + 354.·17-s − 647.·19-s − 1.62e3·20-s − 1.13e3·22-s + 862. i·23-s + ⋯
 L(s)  = 1 + 1.91i·2-s − 2.66·4-s + 1.52·5-s − 0.717·7-s − 3.17i·8-s + 2.92i·10-s + 1.22i·11-s − 0.101i·13-s − 1.37i·14-s + 3.41·16-s + 1.22·17-s − 1.79·19-s − 4.06·20-s − 2.33·22-s + 1.63i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $0.486 + 0.873i$ 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: $$(2,\ 531,\ (\ :2),\ 0.486 + 0.873i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.6325551513$$ $$L(\frac12)$$ $$\approx$$ $$0.6325551513$$ $$L(3)$$ 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 + (1.69e3 + 3.04e3i)T$$
good2 $$1 - 7.65iT - 16T^{2}$$
5 $$1 - 38.1T + 625T^{2}$$
7 $$1 + 35.1T + 2.40e3T^{2}$$
11 $$1 - 147. iT - 1.46e4T^{2}$$
13 $$1 + 17.0iT - 2.85e4T^{2}$$
17 $$1 - 354.T + 8.35e4T^{2}$$
19 $$1 + 647.T + 1.30e5T^{2}$$
23 $$1 - 862. iT - 2.79e5T^{2}$$
29 $$1 + 440.T + 7.07e5T^{2}$$
31 $$1 - 347. iT - 9.23e5T^{2}$$
37 $$1 - 972. iT - 1.87e6T^{2}$$
41 $$1 + 2.78e3T + 2.82e6T^{2}$$
43 $$1 + 2.83e3iT - 3.41e6T^{2}$$
47 $$1 + 1.28e3iT - 4.87e6T^{2}$$
53 $$1 - 1.10e3T + 7.89e6T^{2}$$
61 $$1 + 5.98e3iT - 1.38e7T^{2}$$
67 $$1 - 3.35e3iT - 2.01e7T^{2}$$
71 $$1 - 1.39e3T + 2.54e7T^{2}$$
73 $$1 + 1.64e3iT - 2.83e7T^{2}$$
79 $$1 + 8.35e3T + 3.89e7T^{2}$$
83 $$1 - 5.02e3iT - 4.74e7T^{2}$$
89 $$1 + 5.29e3iT - 6.27e7T^{2}$$
97 $$1 - 1.40e4iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$