# Properties

 Degree $2$ Conductor $800$ Sign $0.894 + 0.447i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 13.4i·3-s + 138. i·7-s + 62.9·9-s + 259.·11-s − 154i·13-s + 178i·17-s − 965.·19-s + 1.86e3·21-s − 2.63e3i·23-s − 4.10e3i·27-s − 4.11e3·29-s + 3.15e3·31-s − 3.48e3i·33-s + 7.44e3i·37-s − 2.06e3·39-s + ⋯
 L(s)  = 1 − 0.860i·3-s + 1.06i·7-s + 0.259·9-s + 0.646·11-s − 0.252i·13-s + 0.149i·17-s − 0.613·19-s + 0.920·21-s − 1.03i·23-s − 1.08i·27-s − 0.907·29-s + 0.590·31-s − 0.556i·33-s + 0.893i·37-s − 0.217·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$800$$    =    $$2^{5} \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Motivic weight: $$5$$ Character: $\chi_{800} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 800,\ (\ :5/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.366805367$$ $$L(\frac12)$$ $$\approx$$ $$2.366805367$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + 13.4iT - 243T^{2}$$
7 $$1 - 138. iT - 1.68e4T^{2}$$
11 $$1 - 259.T + 1.61e5T^{2}$$
13 $$1 + 154iT - 3.71e5T^{2}$$
17 $$1 - 178iT - 1.41e6T^{2}$$
19 $$1 + 965.T + 2.47e6T^{2}$$
23 $$1 + 2.63e3iT - 6.43e6T^{2}$$
29 $$1 + 4.11e3T + 2.05e7T^{2}$$
31 $$1 - 3.15e3T + 2.86e7T^{2}$$
37 $$1 - 7.44e3iT - 6.93e7T^{2}$$
41 $$1 - 7.27e3T + 1.15e8T^{2}$$
43 $$1 - 1.79e4iT - 1.47e8T^{2}$$
47 $$1 + 7.41e3iT - 2.29e8T^{2}$$
53 $$1 + 3.22e4iT - 4.18e8T^{2}$$
59 $$1 + 3.40e4T + 7.14e8T^{2}$$
61 $$1 - 2.67e4T + 8.44e8T^{2}$$
67 $$1 - 4.98e4iT - 1.35e9T^{2}$$
71 $$1 - 5.41e4T + 1.80e9T^{2}$$
73 $$1 - 1.85e4iT - 2.07e9T^{2}$$
79 $$1 - 8.67e4T + 3.07e9T^{2}$$
83 $$1 - 7.86e4iT - 3.93e9T^{2}$$
89 $$1 - 1.07e5T + 5.58e9T^{2}$$
97 $$1 + 1.08e5iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$