# Properties

 Label 2-1280-8.5-c1-0-9 Degree $2$ Conductor $1280$ Sign $0.707 - 0.707i$ Analytic cond. $10.2208$ Root an. cond. $3.19700$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + i·5-s − 4·7-s + 3·9-s − 4i·11-s + 2i·13-s + 2·17-s + 4i·19-s + 4·23-s − 25-s + 2i·29-s + 8·31-s − 4i·35-s + 6i·37-s + 6·41-s + 8i·43-s + ⋯
 L(s)  = 1 + 0.447i·5-s − 1.51·7-s + 9-s − 1.20i·11-s + 0.554i·13-s + 0.485·17-s + 0.917i·19-s + 0.834·23-s − 0.200·25-s + 0.371i·29-s + 1.43·31-s − 0.676i·35-s + 0.986i·37-s + 0.937·41-s + 1.21i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1280$$    =    $$2^{8} \cdot 5$$ Sign: $0.707 - 0.707i$ Analytic conductor: $$10.2208$$ Root analytic conductor: $$3.19700$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1280} (641, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1280,\ (\ :1/2),\ 0.707 - 0.707i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.427581995$$ $$L(\frac12)$$ $$\approx$$ $$1.427581995$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - iT$$
good3 $$1 - 3T^{2}$$
7 $$1 + 4T + 7T^{2}$$
11 $$1 + 4iT - 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
17 $$1 - 2T + 17T^{2}$$
19 $$1 - 4iT - 19T^{2}$$
23 $$1 - 4T + 23T^{2}$$
29 $$1 - 2iT - 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 - 6iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 + 4T + 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 - 4iT - 59T^{2}$$
61 $$1 - 2iT - 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 6T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 16iT - 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 + 14T + 97T^{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

−9.866639121820695831092662345320, −9.118613976944968813447542278707, −8.126689057602863281207603802207, −7.16963968563535773738178697267, −6.42643138239113102869084538153, −5.90031244472374950454902684352, −4.51059440765599261398820940446, −3.48431216676264882025228505285, −2.85511336048940103964150390171, −1.12674084242442291587603208345, 0.72994933058938776222132328125, 2.30841819569299581978980749726, 3.43963126844855277340650502455, 4.42047506778467486043921333025, 5.26677198803139651019079527498, 6.44690641588573265126419483500, 7.02739250018173297176270594124, 7.81045122553104118177974239225, 8.987203995116236834822125741331, 9.735294771613275697138642101526