# Properties

 Degree $2$ Conductor $80$ Sign $-0.804 + 0.593i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 19.8i·3-s + (−45 + 33.1i)5-s + 59.6i·7-s − 153·9-s − 252·11-s − 119. i·13-s + (−660 − 895. i)15-s − 689. i·17-s + 220·19-s − 1.18e3·21-s − 2.43e3i·23-s + (924. − 2.98e3i)25-s + 1.79e3i·27-s − 6.93e3·29-s − 6.75e3·31-s + ⋯
 L(s)  = 1 + 1.27i·3-s + (−0.804 + 0.593i)5-s + 0.460i·7-s − 0.629·9-s − 0.627·11-s − 0.195i·13-s + (−0.757 − 1.02i)15-s − 0.578i·17-s + 0.139·19-s − 0.587·21-s − 0.959i·23-s + (0.295 − 0.955i)25-s + 0.472i·27-s − 1.53·29-s − 1.26·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.804 + 0.593i$ Motivic weight: $$5$$ Character: $\chi_{80} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :5/2),\ -0.804 + 0.593i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.183205 - 0.557366i$$ $$L(\frac12)$$ $$\approx$$ $$0.183205 - 0.557366i$$ $$L(\frac{7}{2})$$ 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 + (45 - 33.1i)T$$
good3 $$1 - 19.8iT - 243T^{2}$$
7 $$1 - 59.6iT - 1.68e4T^{2}$$
11 $$1 + 252T + 1.61e5T^{2}$$
13 $$1 + 119. iT - 3.71e5T^{2}$$
17 $$1 + 689. iT - 1.41e6T^{2}$$
19 $$1 - 220T + 2.47e6T^{2}$$
23 $$1 + 2.43e3iT - 6.43e6T^{2}$$
29 $$1 + 6.93e3T + 2.05e7T^{2}$$
31 $$1 + 6.75e3T + 2.86e7T^{2}$$
37 $$1 - 1.39e4iT - 6.93e7T^{2}$$
41 $$1 + 198T + 1.15e8T^{2}$$
43 $$1 - 417. iT - 1.47e8T^{2}$$
47 $$1 - 1.05e4iT - 2.29e8T^{2}$$
53 $$1 + 5.82e3iT - 4.18e8T^{2}$$
59 $$1 - 2.46e4T + 7.14e8T^{2}$$
61 $$1 + 5.69e3T + 8.44e8T^{2}$$
67 $$1 - 4.36e4iT - 1.35e9T^{2}$$
71 $$1 + 5.33e4T + 1.80e9T^{2}$$
73 $$1 - 7.09e4iT - 2.07e9T^{2}$$
79 $$1 + 5.19e4T + 3.07e9T^{2}$$
83 $$1 - 6.18e4iT - 3.93e9T^{2}$$
89 $$1 + 9.99e3T + 5.58e9T^{2}$$
97 $$1 + 1.01e5iT - 8.58e9T^{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

−14.49752594411039763082254060543, −12.90983674890392720471875134896, −11.58005729088650790866170375174, −10.73355844752777835638069157217, −9.770690845478940023450445049360, −8.540671969408024928511524549821, −7.20400959014858528732836540970, −5.43403235251834930132038685870, −4.16887342677639859435482830729, −2.90215463686548024180531290082, 0.24545829951886585968100040249, 1.72972365910079842681397333888, 3.83913336687543553481393853115, 5.58315848127763899319032472826, 7.25338803591288837699092310252, 7.75938724834058794437686308689, 9.095376398425732577586925992651, 10.81171708991485132610084415289, 11.90777150247344623859242650377, 12.85937452426659599286633648871