# Properties

 Degree $2$ Conductor $320$ Sign $1$ Motivic weight $5$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·3-s − 25·5-s − 192·7-s − 227·9-s − 148·11-s − 286·13-s + 100·15-s − 1.67e3·17-s + 1.06e3·19-s + 768·21-s − 2.97e3·23-s + 625·25-s + 1.88e3·27-s + 3.41e3·29-s + 2.44e3·31-s + 592·33-s + 4.80e3·35-s − 182·37-s + 1.14e3·39-s − 9.39e3·41-s − 1.24e3·43-s + 5.67e3·45-s + 1.20e4·47-s + 2.00e4·49-s + 6.71e3·51-s − 2.38e4·53-s + 3.70e3·55-s + ⋯
 L(s)  = 1 − 0.256·3-s − 0.447·5-s − 1.48·7-s − 0.934·9-s − 0.368·11-s − 0.469·13-s + 0.114·15-s − 1.40·17-s + 0.673·19-s + 0.380·21-s − 1.17·23-s + 1/5·25-s + 0.496·27-s + 0.752·29-s + 0.457·31-s + 0.0946·33-s + 0.662·35-s − 0.0218·37-s + 0.120·39-s − 0.873·41-s − 0.102·43-s + 0.417·45-s + 0.798·47-s + 1.19·49-s + 0.361·51-s − 1.16·53-s + 0.164·55-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $1$ Motivic weight: $$5$$ Character: $\chi_{320} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.4335863759$$ $$L(\frac12)$$ $$\approx$$ $$0.4335863759$$ $$L(\frac{7}{2})$$ 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 + p^{2} T$$
good3 $$1 + 4 T + p^{5} T^{2}$$
7 $$1 + 192 T + p^{5} T^{2}$$
11 $$1 + 148 T + p^{5} T^{2}$$
13 $$1 + 22 p T + p^{5} T^{2}$$
17 $$1 + 1678 T + p^{5} T^{2}$$
19 $$1 - 1060 T + p^{5} T^{2}$$
23 $$1 + 2976 T + p^{5} T^{2}$$
29 $$1 - 3410 T + p^{5} T^{2}$$
31 $$1 - 2448 T + p^{5} T^{2}$$
37 $$1 + 182 T + p^{5} T^{2}$$
41 $$1 + 9398 T + p^{5} T^{2}$$
43 $$1 + 1244 T + p^{5} T^{2}$$
47 $$1 - 12088 T + p^{5} T^{2}$$
53 $$1 + 23846 T + p^{5} T^{2}$$
59 $$1 + 20020 T + p^{5} T^{2}$$
61 $$1 + 32302 T + p^{5} T^{2}$$
67 $$1 - 60972 T + p^{5} T^{2}$$
71 $$1 - 32648 T + p^{5} T^{2}$$
73 $$1 + 38774 T + p^{5} T^{2}$$
79 $$1 - 33360 T + p^{5} T^{2}$$
83 $$1 - 16716 T + p^{5} T^{2}$$
89 $$1 - 101370 T + p^{5} T^{2}$$
97 $$1 + 119038 T + p^{5} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$