# Properties

 Degree $2$ Conductor $80$ Sign $-0.538 - 0.842i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−17.2 + 17.2i)3-s + (46.1 − 31.4i)5-s + (154. + 154. i)7-s − 355. i·9-s − 127. i·11-s + (335. + 335. i)13-s + (−254. + 1.34e3i)15-s + (−1.15e3 + 1.15e3i)17-s + 28.2·19-s − 5.32e3·21-s + (−2.78e3 + 2.78e3i)23-s + (1.14e3 − 2.90e3i)25-s + (1.93e3 + 1.93e3i)27-s + 3.38e3i·29-s + 5.38e3i·31-s + ⋯
 L(s)  = 1 + (−1.10 + 1.10i)3-s + (0.826 − 0.563i)5-s + (1.18 + 1.18i)7-s − 1.46i·9-s − 0.317i·11-s + (0.549 + 0.549i)13-s + (−0.291 + 1.54i)15-s + (−0.969 + 0.969i)17-s + 0.0179·19-s − 2.63·21-s + (−1.09 + 1.09i)23-s + (0.365 − 0.930i)25-s + (0.511 + 0.511i)27-s + 0.748i·29-s + 1.00i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.538 - 0.842i$ Motivic weight: $$5$$ Character: $\chi_{80} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.538 - 0.842i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.648113 + 1.18280i$$ $$L(\frac12)$$ $$\approx$$ $$0.648113 + 1.18280i$$ $$L(\frac{7}{2})$$ 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 + (-46.1 + 31.4i)T$$
good3 $$1 + (17.2 - 17.2i)T - 243iT^{2}$$
7 $$1 + (-154. - 154. i)T + 1.68e4iT^{2}$$
11 $$1 + 127. iT - 1.61e5T^{2}$$
13 $$1 + (-335. - 335. i)T + 3.71e5iT^{2}$$
17 $$1 + (1.15e3 - 1.15e3i)T - 1.41e6iT^{2}$$
19 $$1 - 28.2T + 2.47e6T^{2}$$
23 $$1 + (2.78e3 - 2.78e3i)T - 6.43e6iT^{2}$$
29 $$1 - 3.38e3iT - 2.05e7T^{2}$$
31 $$1 - 5.38e3iT - 2.86e7T^{2}$$
37 $$1 + (-1.15e4 + 1.15e4i)T - 6.93e7iT^{2}$$
41 $$1 + 1.11e4T + 1.15e8T^{2}$$
43 $$1 + (1.43e3 - 1.43e3i)T - 1.47e8iT^{2}$$
47 $$1 + (219. + 219. i)T + 2.29e8iT^{2}$$
53 $$1 + (-2.27e4 - 2.27e4i)T + 4.18e8iT^{2}$$
59 $$1 + 2.21e4T + 7.14e8T^{2}$$
61 $$1 + 1.43e3T + 8.44e8T^{2}$$
67 $$1 + (2.89e4 + 2.89e4i)T + 1.35e9iT^{2}$$
71 $$1 - 2.41e4iT - 1.80e9T^{2}$$
73 $$1 + (-2.85e4 - 2.85e4i)T + 2.07e9iT^{2}$$
79 $$1 - 2.34e4T + 3.07e9T^{2}$$
83 $$1 + (1.89e4 - 1.89e4i)T - 3.93e9iT^{2}$$
89 $$1 + 8.17e3iT - 5.58e9T^{2}$$
97 $$1 + (7.67e4 - 7.67e4i)T - 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$