# Properties

 Degree $2$ Conductor $80$ Sign $0.549 - 0.835i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.36 + 4.55i)2-s + (−18.2 + 18.2i)3-s + (−9.40 − 30.5i)4-s + (−17.6 − 17.6i)5-s + (−21.7 − 144. i)6-s + 13.9i·7-s + (170. + 60.0i)8-s − 424. i·9-s + (139. − 21.0i)10-s + (−220. − 220. i)11-s + (730. + 386. i)12-s + (433. − 433. i)13-s + (−63.3 − 46.7i)14-s + 645.·15-s + (−847. + 575. i)16-s − 25.9·17-s + ⋯
 L(s)  = 1 + (−0.594 + 0.804i)2-s + (−1.17 + 1.17i)3-s + (−0.293 − 0.955i)4-s + (−0.316 − 0.316i)5-s + (−0.246 − 1.63i)6-s + 0.107i·7-s + (0.943 + 0.331i)8-s − 1.74i·9-s + (0.442 − 0.0664i)10-s + (−0.549 − 0.549i)11-s + (1.46 + 0.775i)12-s + (0.711 − 0.711i)13-s + (−0.0863 − 0.0637i)14-s + 0.741·15-s + (−0.827 + 0.561i)16-s − 0.0217·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $0.549 - 0.835i$ Motivic weight: $$5$$ Character: $\chi_{80} (21, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ 0.549 - 0.835i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.526879 + 0.284203i$$ $$L(\frac12)$$ $$\approx$$ $$0.526879 + 0.284203i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (3.36 - 4.55i)T$$
5 $$1 + (17.6 + 17.6i)T$$
good3 $$1 + (18.2 - 18.2i)T - 243iT^{2}$$
7 $$1 - 13.9iT - 1.68e4T^{2}$$
11 $$1 + (220. + 220. i)T + 1.61e5iT^{2}$$
13 $$1 + (-433. + 433. i)T - 3.71e5iT^{2}$$
17 $$1 + 25.9T + 1.41e6T^{2}$$
19 $$1 + (1.56e3 - 1.56e3i)T - 2.47e6iT^{2}$$
23 $$1 - 1.49e3iT - 6.43e6T^{2}$$
29 $$1 + (-324. + 324. i)T - 2.05e7iT^{2}$$
31 $$1 - 7.90e3T + 2.86e7T^{2}$$
37 $$1 + (3.03e3 + 3.03e3i)T + 6.93e7iT^{2}$$
41 $$1 + 1.75e4iT - 1.15e8T^{2}$$
43 $$1 + (-1.12e4 - 1.12e4i)T + 1.47e8iT^{2}$$
47 $$1 - 1.69e4T + 2.29e8T^{2}$$
53 $$1 + (-1.53e4 - 1.53e4i)T + 4.18e8iT^{2}$$
59 $$1 + (2.44e4 + 2.44e4i)T + 7.14e8iT^{2}$$
61 $$1 + (2.89e4 - 2.89e4i)T - 8.44e8iT^{2}$$
67 $$1 + (-4.27e4 + 4.27e4i)T - 1.35e9iT^{2}$$
71 $$1 + 4.05e4iT - 1.80e9T^{2}$$
73 $$1 + 872. iT - 2.07e9T^{2}$$
79 $$1 - 8.71e4T + 3.07e9T^{2}$$
83 $$1 + (6.30e4 - 6.30e4i)T - 3.93e9iT^{2}$$
89 $$1 - 5.13e4iT - 5.58e9T^{2}$$
97 $$1 - 1.17e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$