# Properties

 Label 2-1280-40.27-c1-0-6 Degree $2$ Conductor $1280$ Sign $-0.767 - 0.640i$ 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 + (−1 + 2i)5-s + 3i·9-s + (5 + 5i)13-s + (−5 − 5i)17-s + (−3 − 4i)25-s − 4·29-s + (−5 + 5i)37-s − 8·41-s + (−6 − 3i)45-s + 7i·49-s + (5 + 5i)53-s + 12i·61-s + (−15 + 5i)65-s + (−5 + 5i)73-s − 9·81-s + ⋯
 L(s)  = 1 + (−0.447 + 0.894i)5-s + i·9-s + (1.38 + 1.38i)13-s + (−1.21 − 1.21i)17-s + (−0.600 − 0.800i)25-s − 0.742·29-s + (−0.821 + 0.821i)37-s − 1.24·41-s + (−0.894 − 0.447i)45-s + i·49-s + (0.686 + 0.686i)53-s + 1.53i·61-s + (−1.86 + 0.620i)65-s + (−0.585 + 0.585i)73-s − 81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1280$$    =    $$2^{8} \cdot 5$$ Sign: $-0.767 - 0.640i$ Analytic conductor: $$10.2208$$ Root analytic conductor: $$3.19700$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1280} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1280,\ (\ :1/2),\ -0.767 - 0.640i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.013942835$$ $$L(\frac12)$$ $$\approx$$ $$1.013942835$$ $$L(\frac{3}{2})$$ 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 + (1 - 2i)T$$
good3 $$1 - 3iT^{2}$$
7 $$1 - 7iT^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + (-5 - 5i)T + 13iT^{2}$$
17 $$1 + (5 + 5i)T + 17iT^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 + 4T + 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + (5 - 5i)T - 37iT^{2}$$
41 $$1 + 8T + 41T^{2}$$
43 $$1 - 43iT^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + (-5 - 5i)T + 53iT^{2}$$
59 $$1 - 59T^{2}$$
61 $$1 - 12iT - 61T^{2}$$
67 $$1 + 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (5 - 5i)T - 73iT^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 83iT^{2}$$
89 $$1 + 16iT - 89T^{2}$$
97 $$1 + (-5 - 5i)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

−10.11438865071386317091421550684, −9.030887611588143986310667730503, −8.453517746850719473485622194063, −7.34160995165524630651982063857, −6.85879016846338345739621394763, −5.96700528524974412324000094128, −4.74173148678129836489681767748, −3.97487450572934773574594098011, −2.84378413786553694923164123873, −1.78797119986679657755660265937, 0.42186522704871624138557909427, 1.70153620039890492054305451472, 3.49310370129605724756387465616, 3.91624449322451952782696936069, 5.18109618127051032617823362923, 5.97869096312774114471418214610, 6.79680445208458566379433301339, 7.991150076970423622472806007073, 8.585604864707263456467493059742, 9.091899945531123201644425725516