# Properties

 Label 2-531-59.58-c4-0-78 Degree $2$ Conductor $531$ Sign $-0.736 + 0.676i$ 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 − 4.85i·2-s − 7.60·4-s + 12.5·5-s + 50.4·7-s − 40.7i·8-s − 60.7i·10-s − 191. i·11-s − 0.640i·13-s − 245. i·14-s − 319.·16-s + 549.·17-s + 524.·19-s − 95.0·20-s − 929.·22-s + 543. i·23-s + ⋯
 L(s)  = 1 − 1.21i·2-s − 0.475·4-s + 0.500·5-s + 1.03·7-s − 0.637i·8-s − 0.607i·10-s − 1.58i·11-s − 0.00378i·13-s − 1.25i·14-s − 1.24·16-s + 1.90·17-s + 1.45·19-s − 0.237·20-s − 1.91·22-s + 1.02i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $-0.736 + 0.676i$ 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: $$(2,\ 531,\ (\ :2),\ -0.736 + 0.676i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$3.107006569$$ $$L(\frac12)$$ $$\approx$$ $$3.107006569$$ $$L(3)$$ 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 + (-2.56e3 + 2.35e3i)T$$
good2 $$1 + 4.85iT - 16T^{2}$$
5 $$1 - 12.5T + 625T^{2}$$
7 $$1 - 50.4T + 2.40e3T^{2}$$
11 $$1 + 191. iT - 1.46e4T^{2}$$
13 $$1 + 0.640iT - 2.85e4T^{2}$$
17 $$1 - 549.T + 8.35e4T^{2}$$
19 $$1 - 524.T + 1.30e5T^{2}$$
23 $$1 - 543. iT - 2.79e5T^{2}$$
29 $$1 - 296.T + 7.07e5T^{2}$$
31 $$1 - 1.07e3iT - 9.23e5T^{2}$$
37 $$1 + 1.16e3iT - 1.87e6T^{2}$$
41 $$1 - 1.12e3T + 2.82e6T^{2}$$
43 $$1 - 1.52e3iT - 3.41e6T^{2}$$
47 $$1 + 103. iT - 4.87e6T^{2}$$
53 $$1 - 2.81e3T + 7.89e6T^{2}$$
61 $$1 + 4.90e3iT - 1.38e7T^{2}$$
67 $$1 + 5.85e3iT - 2.01e7T^{2}$$
71 $$1 + 5.82e3T + 2.54e7T^{2}$$
73 $$1 - 6.75e3iT - 2.83e7T^{2}$$
79 $$1 + 9.33e3T + 3.89e7T^{2}$$
83 $$1 + 1.83e3iT - 4.74e7T^{2}$$
89 $$1 + 5.73e3iT - 6.27e7T^{2}$$
97 $$1 - 1.52e4iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$