Properties

 Degree $2$ Conductor $80$ Sign $-0.192 + 0.981i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−38 + 41i)5-s − 243i·9-s + (475 − 475i)13-s + (−1.52e3 − 1.52e3i)17-s + (−237 − 3.11e3i)25-s − 8.56e3i·29-s + (−475 − 475i)37-s − 4.95e3·41-s + (9.96e3 + 9.23e3i)45-s + 1.68e4i·49-s + (−1.64e4 + 1.64e4i)53-s + 5.49e4·61-s + (1.42e3 + 3.75e4i)65-s + (−5.44e4 + 5.44e4i)73-s − 5.90e4·81-s + ⋯
 L(s)  = 1 + (−0.679 + 0.733i)5-s − i·9-s + (0.779 − 0.779i)13-s + (−1.27 − 1.27i)17-s + (−0.0758 − 0.997i)25-s − 1.89i·29-s + (−0.0570 − 0.0570i)37-s − 0.460·41-s + (0.733 + 0.679i)45-s + i·49-s + (−0.805 + 0.805i)53-s + 1.89·61-s + (0.0418 + 1.10i)65-s + (−1.19 + 1.19i)73-s − 0.999·81-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.192 + 0.981i$ Motivic weight: $$5$$ Character: $\chi_{80} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.192 + 0.981i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$0.617059 - 0.749989i$$ $$L(\frac12)$$ $$\approx$$ $$0.617059 - 0.749989i$$ $$L(\frac{7}{2})$$ 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 + (38 - 41i)T$$
good3 $$1 + 243iT^{2}$$
7 $$1 - 1.68e4iT^{2}$$
11 $$1 - 1.61e5T^{2}$$
13 $$1 + (-475 + 475i)T - 3.71e5iT^{2}$$
17 $$1 + (1.52e3 + 1.52e3i)T + 1.41e6iT^{2}$$
19 $$1 + 2.47e6T^{2}$$
23 $$1 + 6.43e6iT^{2}$$
29 $$1 + 8.56e3iT - 2.05e7T^{2}$$
31 $$1 - 2.86e7T^{2}$$
37 $$1 + (475 + 475i)T + 6.93e7iT^{2}$$
41 $$1 + 4.95e3T + 1.15e8T^{2}$$
43 $$1 + 1.47e8iT^{2}$$
47 $$1 - 2.29e8iT^{2}$$
53 $$1 + (1.64e4 - 1.64e4i)T - 4.18e8iT^{2}$$
59 $$1 + 7.14e8T^{2}$$
61 $$1 - 5.49e4T + 8.44e8T^{2}$$
67 $$1 - 1.35e9iT^{2}$$
71 $$1 - 1.80e9T^{2}$$
73 $$1 + (5.44e4 - 5.44e4i)T - 2.07e9iT^{2}$$
79 $$1 + 3.07e9T^{2}$$
83 $$1 + 3.93e9iT^{2}$$
89 $$1 + 1.40e5iT - 5.58e9T^{2}$$
97 $$1 + (1.26e5 + 1.26e5i)T + 8.58e9iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$