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Properties

Label 2-2280-2280.1019-c0-0-1
Degree 22
Conductor 22802280
Sign 0.582−0.813i0.582 - 0.813i
Analytic cond. 1.137861.13786
Root an. cond. 1.066701.06670
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.499i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.499i)20-s + ⋯

Functional equation

Λ(s)=(2280s/2 ΓC(s) L(s)=((0.582−0.813i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2280s/2 ΓC(s) L(s)=((0.582−0.813i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22802280    =    23⋅3⋅5⋅192^{3} \cdot 3 \cdot 5 \cdot 19
Sign: 0.582−0.813i0.582 - 0.813i
Analytic conductor: 1.137861.13786
Root analytic conductor: 1.066701.06670
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2280(1019,⋅)\chi_{2280} (1019, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2280, ( :0), 0.582−0.813i)(2,\ 2280,\ (\ :0),\ 0.582 - 0.813i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.3219063062.321906306
L(12)L(\frac12) ≈\approx 2.3219063062.321906306
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
19 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+T2 1 + T^{2}
11 1−T2 1 - T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
23 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+T+T2 1 + T + T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1+1.73T+T2 1 + 1.73T + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−8.877042733969384921656465505016, −8.370875849070038573106877587971, −7.59933552927184822090306189603, −7.06302164851410156027885233510, −6.43207976545182783963571963334, −5.38184734607222233151067264759, −4.31752805424807140184445634346, −3.48856923427398593848118057442, −3.03275513006769400902655131527, −1.76413024488460118182245408806, 1.27452106443419194414054088915, 2.69366926586840131284632843457, 3.31543636472585341741301699676, 4.19923789622440569034677484028, 4.84861184439535711802950320459, 5.52938692528997917763054119643, 6.91407860474367271981769783210, 7.49525858531761550599893381329, 8.411400507298151637257466083938, 9.187766015056056404791682491396

Graph of the ZZ-function along the critical line