# Properties

 Degree $2$ Conductor $5$ Sign $0.506 + 0.862i$ Motivic weight $8$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.39 − 4.39i)2-s + (75.2 − 75.2i)3-s − 217. i·4-s + (−14.1 + 624. i)5-s − 662.·6-s + (730. + 730. i)7-s + (−2.08e3 + 2.08e3i)8-s − 4.77e3i·9-s + (2.80e3 − 2.68e3i)10-s + 1.95e4·11-s + (−1.63e4 − 1.63e4i)12-s + (−2.49e4 + 2.49e4i)13-s − 6.42e3i·14-s + (4.59e4 + 4.81e4i)15-s − 3.73e4·16-s + (−1.12e4 − 1.12e4i)17-s + ⋯
 L(s)  = 1 + (−0.274 − 0.274i)2-s + (0.929 − 0.929i)3-s − 0.849i·4-s + (−0.0226 + 0.999i)5-s − 0.510·6-s + (0.304 + 0.304i)7-s + (−0.508 + 0.508i)8-s − 0.728i·9-s + (0.280 − 0.268i)10-s + 1.33·11-s + (−0.789 − 0.789i)12-s + (−0.872 + 0.872i)13-s − 0.167i·14-s + (0.908 + 0.950i)15-s − 0.569·16-s + (−0.135 − 0.135i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5$$ Sign: $0.506 + 0.862i$ Motivic weight: $$8$$ Character: $\chi_{5} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5,\ (\ :4),\ 0.506 + 0.862i)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$1.20546 - 0.690118i$$ $$L(\frac12)$$ $$\approx$$ $$1.20546 - 0.690118i$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (14.1 - 624. i)T$$
good2 $$1 + (4.39 + 4.39i)T + 256iT^{2}$$
3 $$1 + (-75.2 + 75.2i)T - 6.56e3iT^{2}$$
7 $$1 + (-730. - 730. i)T + 5.76e6iT^{2}$$
11 $$1 - 1.95e4T + 2.14e8T^{2}$$
13 $$1 + (2.49e4 - 2.49e4i)T - 8.15e8iT^{2}$$
17 $$1 + (1.12e4 + 1.12e4i)T + 6.97e9iT^{2}$$
19 $$1 + 1.71e5iT - 1.69e10T^{2}$$
23 $$1 + (1.32e5 - 1.32e5i)T - 7.83e10iT^{2}$$
29 $$1 + 1.27e5iT - 5.00e11T^{2}$$
31 $$1 + 9.60e5T + 8.52e11T^{2}$$
37 $$1 + (2.43e5 + 2.43e5i)T + 3.51e12iT^{2}$$
41 $$1 - 2.50e6T + 7.98e12T^{2}$$
43 $$1 + (6.76e3 - 6.76e3i)T - 1.16e13iT^{2}$$
47 $$1 + (1.79e6 + 1.79e6i)T + 2.38e13iT^{2}$$
53 $$1 + (2.97e6 - 2.97e6i)T - 6.22e13iT^{2}$$
59 $$1 - 3.13e5iT - 1.46e14T^{2}$$
61 $$1 - 1.76e7T + 1.91e14T^{2}$$
67 $$1 + (-4.41e6 - 4.41e6i)T + 4.06e14iT^{2}$$
71 $$1 + 8.89e6T + 6.45e14T^{2}$$
73 $$1 + (-1.95e7 + 1.95e7i)T - 8.06e14iT^{2}$$
79 $$1 + 1.11e7iT - 1.51e15T^{2}$$
83 $$1 + (-1.58e7 + 1.58e7i)T - 2.25e15iT^{2}$$
89 $$1 - 4.85e7iT - 3.93e15T^{2}$$
97 $$1 + (1.07e8 + 1.07e8i)T + 7.83e15iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$