# Properties

 Degree $2$ Conductor $5$ Sign $0.406 - 0.913i$ Motivic weight $9$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.843i·2-s + 179. i·3-s + 511.·4-s + (−568. + 1.27e3i)5-s − 151.·6-s − 8.71e3i·7-s + 863. i·8-s − 1.24e4·9-s + (−1.07e3 − 479. i)10-s + 4.45e4·11-s + 9.16e4i·12-s − 2.14e4i·13-s + 7.35e3·14-s + (−2.28e5 − 1.01e5i)15-s + 2.61e5·16-s − 3.00e5i·17-s + ⋯
 L(s)  = 1 + 0.0372i·2-s + 1.27i·3-s + 0.998·4-s + (−0.406 + 0.913i)5-s − 0.0476·6-s − 1.37i·7-s + 0.0745i·8-s − 0.632·9-s + (−0.0340 − 0.0151i)10-s + 0.917·11-s + 1.27i·12-s − 0.208i·13-s + 0.0511·14-s + (−1.16 − 0.519i)15-s + 0.995·16-s − 0.871i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5$$ Sign: $0.406 - 0.913i$ Motivic weight: $$9$$ Character: $\chi_{5} (4, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5,\ (\ :9/2),\ 0.406 - 0.913i)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$1.25982 + 0.818238i$$ $$L(\frac12)$$ $$\approx$$ $$1.25982 + 0.818238i$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (568. - 1.27e3i)T$$
good2 $$1 - 0.843iT - 512T^{2}$$
3 $$1 - 179. iT - 1.96e4T^{2}$$
7 $$1 + 8.71e3iT - 4.03e7T^{2}$$
11 $$1 - 4.45e4T + 2.35e9T^{2}$$
13 $$1 + 2.14e4iT - 1.06e10T^{2}$$
17 $$1 + 3.00e5iT - 1.18e11T^{2}$$
19 $$1 + 5.65e5T + 3.22e11T^{2}$$
23 $$1 - 9.50e5iT - 1.80e12T^{2}$$
29 $$1 - 8.03e5T + 1.45e13T^{2}$$
31 $$1 + 1.99e6T + 2.64e13T^{2}$$
37 $$1 + 9.53e6iT - 1.29e14T^{2}$$
41 $$1 + 2.54e7T + 3.27e14T^{2}$$
43 $$1 + 2.32e7iT - 5.02e14T^{2}$$
47 $$1 - 3.77e7iT - 1.11e15T^{2}$$
53 $$1 + 4.79e7iT - 3.29e15T^{2}$$
59 $$1 + 7.00e7T + 8.66e15T^{2}$$
61 $$1 - 1.26e8T + 1.16e16T^{2}$$
67 $$1 - 2.66e8iT - 2.72e16T^{2}$$
71 $$1 - 6.59e7T + 4.58e16T^{2}$$
73 $$1 - 1.47e7iT - 5.88e16T^{2}$$
79 $$1 - 4.66e7T + 1.19e17T^{2}$$
83 $$1 - 2.01e8iT - 1.86e17T^{2}$$
89 $$1 + 5.54e8T + 3.50e17T^{2}$$
97 $$1 + 3.39e8iT - 7.60e17T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$