# Properties

 Degree $2$ Conductor $80$ Sign $0.549 + 0.835i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (9.68 + 9.68i)3-s + (−49.1 − 26.5i)5-s + (48.6 − 48.6i)7-s − 55.4i·9-s − 463. i·11-s + (320. − 320. i)13-s + (−219. − 733. i)15-s + (1.04e3 + 1.04e3i)17-s + 701.·19-s + 942.·21-s + (−2.00e3 − 2.00e3i)23-s + (1.71e3 + 2.61e3i)25-s + (2.89e3 − 2.89e3i)27-s − 3.56e3i·29-s − 9.04e3i·31-s + ⋯
 L(s)  = 1 + (0.621 + 0.621i)3-s + (−0.879 − 0.475i)5-s + (0.375 − 0.375i)7-s − 0.228i·9-s − 1.15i·11-s + (0.526 − 0.526i)13-s + (−0.251 − 0.841i)15-s + (0.877 + 0.877i)17-s + 0.445·19-s + 0.466·21-s + (−0.789 − 0.789i)23-s + (0.548 + 0.836i)25-s + (0.762 − 0.762i)27-s − 0.787i·29-s − 1.69i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $0.549 + 0.835i$ Motivic weight: $$5$$ Character: $\chi_{80} (47, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ 0.549 + 0.835i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.57944 - 0.851321i$$ $$L(\frac12)$$ $$\approx$$ $$1.57944 - 0.851321i$$ $$L(\frac{7}{2})$$ 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 + (49.1 + 26.5i)T$$
good3 $$1 + (-9.68 - 9.68i)T + 243iT^{2}$$
7 $$1 + (-48.6 + 48.6i)T - 1.68e4iT^{2}$$
11 $$1 + 463. iT - 1.61e5T^{2}$$
13 $$1 + (-320. + 320. i)T - 3.71e5iT^{2}$$
17 $$1 + (-1.04e3 - 1.04e3i)T + 1.41e6iT^{2}$$
19 $$1 - 701.T + 2.47e6T^{2}$$
23 $$1 + (2.00e3 + 2.00e3i)T + 6.43e6iT^{2}$$
29 $$1 + 3.56e3iT - 2.05e7T^{2}$$
31 $$1 + 9.04e3iT - 2.86e7T^{2}$$
37 $$1 + (-1.64e3 - 1.64e3i)T + 6.93e7iT^{2}$$
41 $$1 + 1.43e4T + 1.15e8T^{2}$$
43 $$1 + (-3.94e3 - 3.94e3i)T + 1.47e8iT^{2}$$
47 $$1 + (7.94e3 - 7.94e3i)T - 2.29e8iT^{2}$$
53 $$1 + (-1.16e4 + 1.16e4i)T - 4.18e8iT^{2}$$
59 $$1 - 1.12e3T + 7.14e8T^{2}$$
61 $$1 + 2.93e4T + 8.44e8T^{2}$$
67 $$1 + (-9.19e3 + 9.19e3i)T - 1.35e9iT^{2}$$
71 $$1 - 5.26e4iT - 1.80e9T^{2}$$
73 $$1 + (2.79e4 - 2.79e4i)T - 2.07e9iT^{2}$$
79 $$1 - 8.22e4T + 3.07e9T^{2}$$
83 $$1 + (7.72e4 + 7.72e4i)T + 3.93e9iT^{2}$$
89 $$1 - 1.45e5iT - 5.58e9T^{2}$$
97 $$1 + (-9.78e4 - 9.78e4i)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

−13.35267535115370699534100282462, −12.07550005687298249910052822898, −11.03307152421322364429744937521, −9.835522257150010145292328648065, −8.473763672092560895917326102446, −7.942201512877393631302615980625, −5.95651030157787048834987263354, −4.23470592923594339350518334738, −3.33433528116155110741872532270, −0.76124038033039586460059235960, 1.72768310764690907379276228452, 3.28193103182487545986228732021, 4.97956812227033851777444476520, 6.99113388313083275719158999920, 7.69922766881886781064180361428, 8.814434877078632696602391922186, 10.29010214691275498677987680626, 11.65541850907098675874939440363, 12.36005953098642605595706059514, 13.77551564960710290125560042682