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Properties

Label 2-2280-2280.1469-c0-0-1
Degree 22
Conductor 22802280
Sign 0.356−0.934i0.356 - 0.934i
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 + (−0.342 + 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + 0.999i·8-s + (−0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (0.642 + 0.766i)12-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.984 − 0.826i)17-s + (0.984 + 0.173i)18-s + (0.939 + 0.342i)19-s + (−0.342 + 0.939i)20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.342 + 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + 0.999i·8-s + (−0.766 − 0.642i)9-s + (0.766 − 0.642i)10-s + (0.642 + 0.766i)12-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + (0.984 − 0.826i)17-s + (0.984 + 0.173i)18-s + (0.939 + 0.342i)19-s + (−0.342 + 0.939i)20-s + ⋯

Functional equation

Λ(s)=(2280s/2 ΓC(s) L(s)=((0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2280s/2 ΓC(s) L(s)=((0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22802280    =    23⋅3⋅5⋅192^{3} \cdot 3 \cdot 5 \cdot 19
Sign: 0.356−0.934i0.356 - 0.934i
Analytic conductor: 1.137861.13786
Root analytic conductor: 1.066701.06670
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2280(1469,⋅)\chi_{2280} (1469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2280, ( :0), 0.356−0.934i)(2,\ 2280,\ (\ :0),\ 0.356 - 0.934i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.55941107620.5594110762
L(12)L(\frac12) ≈\approx 0.55941107620.5594110762
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+(0.342−0.939i)T 1 + (0.342 - 0.939i)T
5 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
19 1+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
good7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
17 1+(−0.984+0.826i)T+(0.173−0.984i)T2 1 + (-0.984 + 0.826i)T + (0.173 - 0.984i)T^{2}
23 1+(−0.300+1.70i)T+(−0.939−0.342i)T2 1 + (-0.300 + 1.70i)T + (-0.939 - 0.342i)T^{2}
29 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
31 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
43 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
47 1+(−1.50−1.26i)T+(0.173+0.984i)T2 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2}
53 1+(1.50+0.266i)T+(0.939+0.342i)T2 1 + (1.50 + 0.266i)T + (0.939 + 0.342i)T^{2}
59 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
61 1+(−1.70−0.300i)T+(0.939+0.342i)T2 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2}
67 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
79 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
83 1+(−1.62−0.939i)T+(0.5+0.866i)T2 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2}
89 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
97 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)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

−9.304573108382036298277292691030, −8.626954745748782231830267109114, −7.903597395325695761124171059165, −7.12306744235645760941316032433, −6.39527866384974187456422957172, −5.31503560458412692321900782641, −4.82358298734764377092691701561, −3.62479648052862208774986496355, −2.77248744371716278099617990550, −0.840994869105297328409824473227, 0.837298822527157808663962189938, 1.87670867168837745676451316214, 3.15654836774042567337091871207, 3.83735389959299968076707472640, 5.21316917261079897811744817207, 6.09238518292269573647913056621, 7.20553568121515245374868377115, 7.60352099340058831011902083298, 8.086682864780731140491821048420, 9.010646330433010551317661808961

Graph of the ZZ-function along the critical line