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Properties

Label 2-280-280.179-c0-0-0
Degree $2$
Conductor $280$
Sign $0.0633 - 0.997i$
Analytic cond. $0.139738$
Root an. cond. $0.373815$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 7-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s − 0.999·20-s + 0.999·22-s + (−0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 7-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s − 0.999·20-s + 0.999·22-s + (−0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(0.139738\)
Root analytic conductor: \(0.373815\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :0),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9003560449\)
\(L(\frac12)\) \(\approx\) \(0.9003560449\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.43661702903630611698648175216, −11.58228246400703646156965992564, −10.34981008285597145419030574862, −9.259868635247810357625713530602, −8.494984857385028792438112678468, −7.08164090917019720194429738367, −6.14344346913775295094976015169, −5.88650311431721599084779389603, −3.76275309843798213587506784737, −3.13096812951106527186115780787, 1.75406181552643421361357220844, 3.27215264224217588357928356523, 4.61786745847556869163907995676, 5.57308134796154957948202176018, 6.60667523442364763206691892267, 8.340390900610433125282624132098, 9.367759933787970351658553263718, 9.899670395063341170480745356369, 11.05087231705270468550520864070, 11.95801266419308080269532549316

Graph of the $Z$-function along the critical line