# 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 + 21.6i·3-s − 25i·5-s − 169.·7-s − 224.·9-s − 512. i·11-s − 36.4i·13-s + 540.·15-s − 1.26e3·17-s + 790. i·19-s − 3.66e3i·21-s + 4.99e3·23-s − 625·25-s + 401. i·27-s + 934. i·29-s + 6.69e3·31-s + ⋯
 L(s)  = 1 + 1.38i·3-s − 0.447i·5-s − 1.30·7-s − 0.923·9-s − 1.27i·11-s − 0.0598i·13-s + 0.620·15-s − 1.05·17-s + 0.502i·19-s − 1.81i·21-s + 1.96·23-s − 0.200·25-s + 0.106i·27-s + 0.206i·29-s + 1.25·31-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

 $$L(3)$$ $$\approx$$ $$1.483656199$$ $$L(\frac12)$$ $$\approx$$ $$1.483656199$$ $$L(\frac{7}{2})$$ 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 - 21.6iT - 243T^{2}$$
7 $$1 + 169.T + 1.68e4T^{2}$$
11 $$1 + 512. iT - 1.61e5T^{2}$$
13 $$1 + 36.4iT - 3.71e5T^{2}$$
17 $$1 + 1.26e3T + 1.41e6T^{2}$$
19 $$1 - 790. iT - 2.47e6T^{2}$$
23 $$1 - 4.99e3T + 6.43e6T^{2}$$
29 $$1 - 934. iT - 2.05e7T^{2}$$
31 $$1 - 6.69e3T + 2.86e7T^{2}$$
37 $$1 + 4.30e3iT - 6.93e7T^{2}$$
41 $$1 + 1.03e4T + 1.15e8T^{2}$$
43 $$1 - 6.37e3iT - 1.47e8T^{2}$$
47 $$1 - 2.21e4T + 2.29e8T^{2}$$
53 $$1 + 2.32e4iT - 4.18e8T^{2}$$
59 $$1 + 2.77e4iT - 7.14e8T^{2}$$
61 $$1 - 3.99e4iT - 8.44e8T^{2}$$
67 $$1 - 5.54e4iT - 1.35e9T^{2}$$
71 $$1 - 3.98e4T + 1.80e9T^{2}$$
73 $$1 + 435.T + 2.07e9T^{2}$$
79 $$1 + 6.16e4T + 3.07e9T^{2}$$
83 $$1 - 2.94e3iT - 3.93e9T^{2}$$
89 $$1 - 6.47e4T + 5.58e9T^{2}$$
97 $$1 - 1.59e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$