Properties

 Label 2-531-59.58-c4-0-2 Degree $2$ Conductor $531$ Sign $-0.00986 - 0.999i$ 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 − 2.33i·2-s + 10.5·4-s − 30.0·5-s − 41.7·7-s − 62.0i·8-s + 70.2i·10-s − 117. i·11-s − 118. i·13-s + 97.6i·14-s + 23.2·16-s − 263.·17-s + 373.·19-s − 316.·20-s − 274.·22-s − 503. i·23-s + ⋯
 L(s)  = 1 − 0.584i·2-s + 0.657·4-s − 1.20·5-s − 0.851·7-s − 0.969i·8-s + 0.702i·10-s − 0.970i·11-s − 0.700i·13-s + 0.498i·14-s + 0.0906·16-s − 0.913·17-s + 1.03·19-s − 0.790·20-s − 0.567·22-s − 0.951i·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00986 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00986 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$531$$    =    $$3^{2} \cdot 59$$ Sign: $-0.00986 - 0.999i$ 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: $$(2,\ 531,\ (\ :2),\ -0.00986 - 0.999i)$$

Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$0.07955323401$$ $$L(\frac12)$$ $$\approx$$ $$0.07955323401$$ $$L(3)$$ 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 + (-34.3 - 3.48e3i)T$$
good2 $$1 + 2.33iT - 16T^{2}$$
5 $$1 + 30.0T + 625T^{2}$$
7 $$1 + 41.7T + 2.40e3T^{2}$$
11 $$1 + 117. iT - 1.46e4T^{2}$$
13 $$1 + 118. iT - 2.85e4T^{2}$$
17 $$1 + 263.T + 8.35e4T^{2}$$
19 $$1 - 373.T + 1.30e5T^{2}$$
23 $$1 + 503. iT - 2.79e5T^{2}$$
29 $$1 - 549.T + 7.07e5T^{2}$$
31 $$1 - 450. iT - 9.23e5T^{2}$$
37 $$1 - 1.43e3iT - 1.87e6T^{2}$$
41 $$1 + 1.44e3T + 2.82e6T^{2}$$
43 $$1 - 3.20e3iT - 3.41e6T^{2}$$
47 $$1 - 967. iT - 4.87e6T^{2}$$
53 $$1 + 1.66e3T + 7.89e6T^{2}$$
61 $$1 + 1.71e3iT - 1.38e7T^{2}$$
67 $$1 - 5.06e3iT - 2.01e7T^{2}$$
71 $$1 + 6.15e3T + 2.54e7T^{2}$$
73 $$1 + 4.17e3iT - 2.83e7T^{2}$$
79 $$1 + 5.99e3T + 3.89e7T^{2}$$
83 $$1 + 2.93e3iT - 4.74e7T^{2}$$
89 $$1 + 3.87e3iT - 6.27e7T^{2}$$
97 $$1 - 2.34e3iT - 8.85e7T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$