# Properties

 Degree $2$ Conductor $80$ Sign $-0.581 + 0.813i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (20.3 − 20.3i)3-s + (−46.4 − 31.1i)5-s + (76.9 + 76.9i)7-s − 588. i·9-s − 556. i·11-s + (141. + 141. i)13-s + (−1.58e3 + 312. i)15-s + (−477. + 477. i)17-s − 1.60e3·19-s + 3.13e3·21-s + (346. − 346. i)23-s + (1.19e3 + 2.88e3i)25-s + (−7.03e3 − 7.03e3i)27-s − 7.48e3i·29-s + 7.92e3i·31-s + ⋯
 L(s)  = 1 + (1.30 − 1.30i)3-s + (−0.830 − 0.556i)5-s + (0.593 + 0.593i)7-s − 2.41i·9-s − 1.38i·11-s + (0.231 + 0.231i)13-s + (−1.81 + 0.359i)15-s + (−0.400 + 0.400i)17-s − 1.02·19-s + 1.55·21-s + (0.136 − 0.136i)23-s + (0.380 + 0.924i)25-s + (−1.85 − 1.85i)27-s − 1.65i·29-s + 1.48i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.581 + 0.813i$ Motivic weight: $$5$$ Character: $\chi_{80} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.581 + 0.813i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.04784 - 2.03553i$$ $$L(\frac12)$$ $$\approx$$ $$1.04784 - 2.03553i$$ $$L(\frac{7}{2})$$ 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 + (46.4 + 31.1i)T$$
good3 $$1 + (-20.3 + 20.3i)T - 243iT^{2}$$
7 $$1 + (-76.9 - 76.9i)T + 1.68e4iT^{2}$$
11 $$1 + 556. iT - 1.61e5T^{2}$$
13 $$1 + (-141. - 141. i)T + 3.71e5iT^{2}$$
17 $$1 + (477. - 477. i)T - 1.41e6iT^{2}$$
19 $$1 + 1.60e3T + 2.47e6T^{2}$$
23 $$1 + (-346. + 346. i)T - 6.43e6iT^{2}$$
29 $$1 + 7.48e3iT - 2.05e7T^{2}$$
31 $$1 - 7.92e3iT - 2.86e7T^{2}$$
37 $$1 + (-3.32e3 + 3.32e3i)T - 6.93e7iT^{2}$$
41 $$1 - 1.87e4T + 1.15e8T^{2}$$
43 $$1 + (-8.25e3 + 8.25e3i)T - 1.47e8iT^{2}$$
47 $$1 + (-5.09e3 - 5.09e3i)T + 2.29e8iT^{2}$$
53 $$1 + (-1.94e4 - 1.94e4i)T + 4.18e8iT^{2}$$
59 $$1 + 108.T + 7.14e8T^{2}$$
61 $$1 - 1.42e4T + 8.44e8T^{2}$$
67 $$1 + (-2.89e4 - 2.89e4i)T + 1.35e9iT^{2}$$
71 $$1 + 982. iT - 1.80e9T^{2}$$
73 $$1 + (2.15e4 + 2.15e4i)T + 2.07e9iT^{2}$$
79 $$1 - 9.38e3T + 3.07e9T^{2}$$
83 $$1 + (9.45e3 - 9.45e3i)T - 3.93e9iT^{2}$$
89 $$1 - 8.48e3iT - 5.58e9T^{2}$$
97 $$1 + (-1.22e5 + 1.22e5i)T - 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$