# Properties

 Degree $2$ Conductor $320$ Sign $0.707 + 0.707i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 9.72i·3-s + 25i·5-s + 65.1·7-s + 148.·9-s − 71.5i·11-s − 733. i·13-s − 243.·15-s − 716.·17-s − 2.33e3i·19-s + 633. i·21-s − 278.·23-s − 625·25-s + 3.80e3i·27-s − 2.95e3i·29-s − 6.04e3·31-s + ⋯
 L(s)  = 1 + 0.623i·3-s + 0.447i·5-s + 0.502·7-s + 0.610·9-s − 0.178i·11-s − 1.20i·13-s − 0.279·15-s − 0.600·17-s − 1.48i·19-s + 0.313i·21-s − 0.109·23-s − 0.200·25-s + 1.00i·27-s − 0.652i·29-s − 1.13·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $0.707 + 0.707i$ Motivic weight: $$5$$ Character: $\chi_{320} (161, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :5/2),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.842657892$$ $$L(\frac12)$$ $$\approx$$ $$1.842657892$$ $$L(\frac{7}{2})$$ 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 - 25iT$$
good3 $$1 - 9.72iT - 243T^{2}$$
7 $$1 - 65.1T + 1.68e4T^{2}$$
11 $$1 + 71.5iT - 1.61e5T^{2}$$
13 $$1 + 733. iT - 3.71e5T^{2}$$
17 $$1 + 716.T + 1.41e6T^{2}$$
19 $$1 + 2.33e3iT - 2.47e6T^{2}$$
23 $$1 + 278.T + 6.43e6T^{2}$$
29 $$1 + 2.95e3iT - 2.05e7T^{2}$$
31 $$1 + 6.04e3T + 2.86e7T^{2}$$
37 $$1 + 1.21e4iT - 6.93e7T^{2}$$
41 $$1 - 3.62e3T + 1.15e8T^{2}$$
43 $$1 - 230. iT - 1.47e8T^{2}$$
47 $$1 - 1.39e4T + 2.29e8T^{2}$$
53 $$1 + 2.24e4iT - 4.18e8T^{2}$$
59 $$1 - 7.28e3iT - 7.14e8T^{2}$$
61 $$1 + 1.94e4iT - 8.44e8T^{2}$$
67 $$1 - 1.39e3iT - 1.35e9T^{2}$$
71 $$1 - 5.73e4T + 1.80e9T^{2}$$
73 $$1 + 3.55e4T + 2.07e9T^{2}$$
79 $$1 - 9.30e4T + 3.07e9T^{2}$$
83 $$1 - 1.59e3iT - 3.93e9T^{2}$$
89 $$1 - 2.02e3T + 5.58e9T^{2}$$
97 $$1 + 1.20e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$