# Properties

 Label 2-80-80.67-c1-0-0 Degree $2$ Conductor $80$ Sign $-0.637 - 0.770i$ Analytic cond. $0.638803$ Root an. cond. $0.799251$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.34 + 0.430i)2-s + 2.96i·3-s + (1.62 − 1.15i)4-s + (−2.22 + 0.177i)5-s + (−1.27 − 3.99i)6-s + (−0.115 − 0.115i)7-s + (−1.69 + 2.26i)8-s − 5.79·9-s + (2.92 − 1.19i)10-s + (2.95 + 2.95i)11-s + (3.43 + 4.83i)12-s + 1.55·13-s + (0.204 + 0.105i)14-s + (−0.525 − 6.61i)15-s + (1.31 − 3.77i)16-s + (0.299 + 0.299i)17-s + ⋯
 L(s)  = 1 + (−0.952 + 0.304i)2-s + 1.71i·3-s + (0.814 − 0.579i)4-s + (−0.996 + 0.0793i)5-s + (−0.520 − 1.63i)6-s + (−0.0435 − 0.0435i)7-s + (−0.599 + 0.800i)8-s − 1.93·9-s + (0.925 − 0.378i)10-s + (0.892 + 0.892i)11-s + (0.992 + 1.39i)12-s + 0.432·13-s + (0.0546 + 0.0282i)14-s + (−0.135 − 1.70i)15-s + (0.327 − 0.944i)16-s + (0.0726 + 0.0726i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.637 - 0.770i$ Analytic conductor: $$0.638803$$ Root analytic conductor: $$0.799251$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{80} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :1/2),\ -0.637 - 0.770i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.235694 + 0.501155i$$ $$L(\frac12)$$ $$\approx$$ $$0.235694 + 0.501155i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.34 - 0.430i)T$$
5 $$1 + (2.22 - 0.177i)T$$
good3 $$1 - 2.96iT - 3T^{2}$$
7 $$1 + (0.115 + 0.115i)T + 7iT^{2}$$
11 $$1 + (-2.95 - 2.95i)T + 11iT^{2}$$
13 $$1 - 1.55T + 13T^{2}$$
17 $$1 + (-0.299 - 0.299i)T + 17iT^{2}$$
19 $$1 + (-2.26 - 2.26i)T + 19iT^{2}$$
23 $$1 + (-4.14 + 4.14i)T - 23iT^{2}$$
29 $$1 + (0.289 - 0.289i)T - 29iT^{2}$$
31 $$1 - 4.18iT - 31T^{2}$$
37 $$1 - 1.63T + 37T^{2}$$
41 $$1 + 7.61iT - 41T^{2}$$
43 $$1 + 6.72T + 43T^{2}$$
47 $$1 + (4.38 - 4.38i)T - 47iT^{2}$$
53 $$1 + 11.4iT - 53T^{2}$$
59 $$1 + (1.63 - 1.63i)T - 59iT^{2}$$
61 $$1 + (1.23 + 1.23i)T + 61iT^{2}$$
67 $$1 - 2.49T + 67T^{2}$$
71 $$1 - 8.00T + 71T^{2}$$
73 $$1 + (-1.12 - 1.12i)T + 73iT^{2}$$
79 $$1 + 3.62T + 79T^{2}$$
83 $$1 + 1.62iT - 83T^{2}$$
89 $$1 - 15.7T + 89T^{2}$$
97 $$1 + (-9.69 - 9.69i)T + 97iT^{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

−14.99365918692311576816285144057, −14.54888207667535586665276584678, −12.07402864300686661499991046723, −11.12506013150074619072091714100, −10.24018401830656735557880392071, −9.260697758712646284264909844080, −8.324141314784163581911549430843, −6.79520217848282848916893588895, −4.97531332359176309055620556335, −3.56168342163255562777626767352, 1.07650061040305668188564730603, 3.20186047640124971935993350644, 6.28337041235252234115941030529, 7.29789643498546569957732131233, 8.159254017763511876507788307744, 9.124581514170267242516462900865, 11.28303940405842313284548993934, 11.60282036406157116285470985844, 12.69088125615007132745178510924, 13.67457800793446000096858793390