# Properties

 Degree $2$ Conductor $80$ Sign $-0.419 - 0.907i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.69i·3-s + (23.4 + 50.7i)5-s − 10.2i·7-s + 220.·9-s − 596.·11-s + 420. i·13-s + (−238. + 109. i)15-s + 974. i·17-s − 380.·19-s + 48.1·21-s + 3.54e3i·23-s + (−2.02e3 + 2.37e3i)25-s + 2.17e3i·27-s − 5.44e3·29-s + 3.62e3·31-s + ⋯
 L(s)  = 1 + 0.301i·3-s + (0.419 + 0.907i)5-s − 0.0791i·7-s + 0.909·9-s − 1.48·11-s + 0.690i·13-s + (−0.273 + 0.126i)15-s + 0.817i·17-s − 0.241·19-s + 0.0238·21-s + 1.39i·23-s + (−0.648 + 0.760i)25-s + 0.574i·27-s − 1.20·29-s + 0.677·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.419 - 0.907i$ Motivic weight: $$5$$ Character: $\chi_{80} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.419 - 0.907i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.817024 + 1.27687i$$ $$L(\frac12)$$ $$\approx$$ $$0.817024 + 1.27687i$$ $$L(\frac{7}{2})$$ 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 + (-23.4 - 50.7i)T$$
good3 $$1 - 4.69iT - 243T^{2}$$
7 $$1 + 10.2iT - 1.68e4T^{2}$$
11 $$1 + 596.T + 1.61e5T^{2}$$
13 $$1 - 420. iT - 3.71e5T^{2}$$
17 $$1 - 974. iT - 1.41e6T^{2}$$
19 $$1 + 380.T + 2.47e6T^{2}$$
23 $$1 - 3.54e3iT - 6.43e6T^{2}$$
29 $$1 + 5.44e3T + 2.05e7T^{2}$$
31 $$1 - 3.62e3T + 2.86e7T^{2}$$
37 $$1 - 1.75e3iT - 6.93e7T^{2}$$
41 $$1 - 263.T + 1.15e8T^{2}$$
43 $$1 - 1.44e4iT - 1.47e8T^{2}$$
47 $$1 + 2.34e4iT - 2.29e8T^{2}$$
53 $$1 + 3.34e4iT - 4.18e8T^{2}$$
59 $$1 + 2.90e3T + 7.14e8T^{2}$$
61 $$1 - 2.94e4T + 8.44e8T^{2}$$
67 $$1 + 7.16e3iT - 1.35e9T^{2}$$
71 $$1 - 8.13e4T + 1.80e9T^{2}$$
73 $$1 + 5.51e4iT - 2.07e9T^{2}$$
79 $$1 - 1.64e4T + 3.07e9T^{2}$$
83 $$1 - 1.16e5iT - 3.93e9T^{2}$$
89 $$1 + 9.93e4T + 5.58e9T^{2}$$
97 $$1 - 6.29e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$