# Properties

 Degree $2$ Conductor $80$ Sign $-0.918 - 0.395i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−7.48 + 7.48i)3-s + (34.7 + 43.8i)5-s + (−19.2 − 19.2i)7-s + 131. i·9-s − 180. i·11-s + (44.2 + 44.2i)13-s + (−587. − 67.9i)15-s + (−621. + 621. i)17-s − 2.67e3·19-s + 287.·21-s + (−2.23e3 + 2.23e3i)23-s + (−712. + 3.04e3i)25-s + (−2.79e3 − 2.79e3i)27-s + 705. i·29-s − 2.76e3i·31-s + ⋯
 L(s)  = 1 + (−0.480 + 0.480i)3-s + (0.621 + 0.783i)5-s + (−0.148 − 0.148i)7-s + 0.539i·9-s − 0.450i·11-s + (0.0725 + 0.0725i)13-s + (−0.674 − 0.0779i)15-s + (−0.521 + 0.521i)17-s − 1.69·19-s + 0.142·21-s + (−0.880 + 0.880i)23-s + (−0.228 + 0.973i)25-s + (−0.738 − 0.738i)27-s + 0.155i·29-s − 0.516i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.918 - 0.395i$ Motivic weight: $$5$$ Character: $\chi_{80} (63, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.918 - 0.395i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.185648 + 0.900081i$$ $$L(\frac12)$$ $$\approx$$ $$0.185648 + 0.900081i$$ $$L(\frac{7}{2})$$ 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 + (-34.7 - 43.8i)T$$
good3 $$1 + (7.48 - 7.48i)T - 243iT^{2}$$
7 $$1 + (19.2 + 19.2i)T + 1.68e4iT^{2}$$
11 $$1 + 180. iT - 1.61e5T^{2}$$
13 $$1 + (-44.2 - 44.2i)T + 3.71e5iT^{2}$$
17 $$1 + (621. - 621. i)T - 1.41e6iT^{2}$$
19 $$1 + 2.67e3T + 2.47e6T^{2}$$
23 $$1 + (2.23e3 - 2.23e3i)T - 6.43e6iT^{2}$$
29 $$1 - 705. iT - 2.05e7T^{2}$$
31 $$1 + 2.76e3iT - 2.86e7T^{2}$$
37 $$1 + (3.54e3 - 3.54e3i)T - 6.93e7iT^{2}$$
41 $$1 - 1.09e4T + 1.15e8T^{2}$$
43 $$1 + (5.34e3 - 5.34e3i)T - 1.47e8iT^{2}$$
47 $$1 + (-1.32e4 - 1.32e4i)T + 2.29e8iT^{2}$$
53 $$1 + (1.56e4 + 1.56e4i)T + 4.18e8iT^{2}$$
59 $$1 - 4.59e4T + 7.14e8T^{2}$$
61 $$1 + 1.75e4T + 8.44e8T^{2}$$
67 $$1 + (-3.10e4 - 3.10e4i)T + 1.35e9iT^{2}$$
71 $$1 + 1.06e4iT - 1.80e9T^{2}$$
73 $$1 + (-5.09e4 - 5.09e4i)T + 2.07e9iT^{2}$$
79 $$1 - 2.47e4T + 3.07e9T^{2}$$
83 $$1 + (4.89e4 - 4.89e4i)T - 3.93e9iT^{2}$$
89 $$1 - 2.61e4iT - 5.58e9T^{2}$$
97 $$1 + (-3.99e4 + 3.99e4i)T - 8.58e9iT^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.86138263444926483173789611741, −12.96011722767991110197591705185, −11.31925719912318351091439777819, −10.65343543907011375453309598564, −9.722425993199205011482825074467, −8.203084983500022046479385705266, −6.63045440539900530615547518668, −5.62173652057945035713717491240, −4.01582528470056271739568467321, −2.18052429341612950612465615346, 0.39061172290722094274041907018, 2.06616571229224580739448636445, 4.37438440634257755749509176793, 5.84477399037664692774621219795, 6.79422534177500689430566618522, 8.481244384268403838663912683460, 9.484052425060715621296486749395, 10.75481187271266998048408458589, 12.24813086326390261440691093032, 12.65908039940480201936565805595