# Properties

 Label 2-531-59.58-c4-0-39 Degree $2$ Conductor $531$ Sign $-0.444 + 0.895i$ 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 − 5.77i·2-s − 17.3·4-s − 34.3·5-s − 39.2·7-s + 7.56i·8-s + 198. i·10-s + 56.8i·11-s + 232. i·13-s + 226. i·14-s − 233.·16-s + 391.·17-s − 352.·19-s + 594.·20-s + 328.·22-s − 173. i·23-s + ⋯
 L(s)  = 1 − 1.44i·2-s − 1.08·4-s − 1.37·5-s − 0.801·7-s + 0.118i·8-s + 1.98i·10-s + 0.469i·11-s + 1.37i·13-s + 1.15i·14-s − 0.911·16-s + 1.35·17-s − 0.976·19-s + 1.48·20-s + 0.677·22-s − 0.327i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $-0.444 + 0.895i$ 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: $$(2,\ 531,\ (\ :2),\ -0.444 + 0.895i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.8653730655$$ $$L(\frac12)$$ $$\approx$$ $$0.8653730655$$ $$L(3)$$ 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 + (-1.54e3 + 3.11e3i)T$$
good2 $$1 + 5.77iT - 16T^{2}$$
5 $$1 + 34.3T + 625T^{2}$$
7 $$1 + 39.2T + 2.40e3T^{2}$$
11 $$1 - 56.8iT - 1.46e4T^{2}$$
13 $$1 - 232. iT - 2.85e4T^{2}$$
17 $$1 - 391.T + 8.35e4T^{2}$$
19 $$1 + 352.T + 1.30e5T^{2}$$
23 $$1 + 173. iT - 2.79e5T^{2}$$
29 $$1 + 90.7T + 7.07e5T^{2}$$
31 $$1 + 128. iT - 9.23e5T^{2}$$
37 $$1 - 413. iT - 1.87e6T^{2}$$
41 $$1 - 761.T + 2.82e6T^{2}$$
43 $$1 + 874. iT - 3.41e6T^{2}$$
47 $$1 - 895. iT - 4.87e6T^{2}$$
53 $$1 + 3.68e3T + 7.89e6T^{2}$$
61 $$1 + 1.49e3iT - 1.38e7T^{2}$$
67 $$1 - 3.68e3iT - 2.01e7T^{2}$$
71 $$1 - 9.23e3T + 2.54e7T^{2}$$
73 $$1 + 4.30e3iT - 2.83e7T^{2}$$
79 $$1 + 4.70e3T + 3.89e7T^{2}$$
83 $$1 + 398. iT - 4.74e7T^{2}$$
89 $$1 + 1.24e4iT - 6.27e7T^{2}$$
97 $$1 + 397. iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$