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Properties

Label 2-990-11.4-c1-0-5
Degree 22
Conductor 990990
Sign −0.394−0.918i-0.394 - 0.918i
Analytic cond. 7.905187.90518
Root an. cond. 2.811612.81161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.809 + 2.48i)7-s + (−0.309 + 0.951i)8-s − 10-s + (2.54 − 2.12i)11-s + (1.30 + 0.951i)13-s + (−0.809 + 2.48i)14-s + (−0.809 + 0.587i)16-s + (−4.23 + 3.07i)17-s + (−1.26 + 3.88i)19-s + (−0.809 − 0.587i)20-s + (3.30 − 0.224i)22-s − 0.145·23-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (0.305 + 0.941i)7-s + (−0.109 + 0.336i)8-s − 0.316·10-s + (0.767 − 0.641i)11-s + (0.363 + 0.263i)13-s + (−0.216 + 0.665i)14-s + (−0.202 + 0.146i)16-s + (−1.02 + 0.746i)17-s + (−0.289 + 0.892i)19-s + (−0.180 − 0.131i)20-s + (0.705 − 0.0478i)22-s − 0.0304·23-s + ⋯

Functional equation

Λ(s)=(990s/2 ΓC(s) L(s)=((−0.394−0.918i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(990s/2 ΓC(s+1/2) L(s)=((−0.394−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 990990    =    2⋅32⋅5⋅112 \cdot 3^{2} \cdot 5 \cdot 11
Sign: −0.394−0.918i-0.394 - 0.918i
Analytic conductor: 7.905187.90518
Root analytic conductor: 2.811612.81161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ990(631,⋅)\chi_{990} (631, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 990, ( :1/2), −0.394−0.918i)(2,\ 990,\ (\ :1/2),\ -0.394 - 0.918i)

Particular Values

L(1)L(1) ≈\approx 1.08920+1.65324i1.08920 + 1.65324i
L(12)L(\frac12) ≈\approx 1.08920+1.65324i1.08920 + 1.65324i
L(32)L(\frac{3}{2}) 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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1 1
5 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1+(−2.54+2.12i)T 1 + (-2.54 + 2.12i)T
good7 1+(−0.809−2.48i)T+(−5.66+4.11i)T2 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.30−0.951i)T+(4.01+12.3i)T2 1 + (-1.30 - 0.951i)T + (4.01 + 12.3i)T^{2}
17 1+(4.23−3.07i)T+(5.25−16.1i)T2 1 + (4.23 - 3.07i)T + (5.25 - 16.1i)T^{2}
19 1+(1.26−3.88i)T+(−15.3−11.1i)T2 1 + (1.26 - 3.88i)T + (-15.3 - 11.1i)T^{2}
23 1+0.145T+23T2 1 + 0.145T + 23T^{2}
29 1+(−0.381−1.17i)T+(−23.4+17.0i)T2 1 + (-0.381 - 1.17i)T + (-23.4 + 17.0i)T^{2}
31 1+(−5.85−4.25i)T+(9.57+29.4i)T2 1 + (-5.85 - 4.25i)T + (9.57 + 29.4i)T^{2}
37 1+(0.263+0.812i)T+(−29.9+21.7i)T2 1 + (0.263 + 0.812i)T + (-29.9 + 21.7i)T^{2}
41 1+(−0.572+1.76i)T+(−33.1−24.0i)T2 1 + (-0.572 + 1.76i)T + (-33.1 - 24.0i)T^{2}
43 1+9.23T+43T2 1 + 9.23T + 43T^{2}
47 1+(3.5−10.7i)T+(−38.0−27.6i)T2 1 + (3.5 - 10.7i)T + (-38.0 - 27.6i)T^{2}
53 1+(−0.736−0.534i)T+(16.3+50.4i)T2 1 + (-0.736 - 0.534i)T + (16.3 + 50.4i)T^{2}
59 1+(0.736+2.26i)T+(−47.7+34.6i)T2 1 + (0.736 + 2.26i)T + (-47.7 + 34.6i)T^{2}
61 1+(7.23−5.25i)T+(18.8−58.0i)T2 1 + (7.23 - 5.25i)T + (18.8 - 58.0i)T^{2}
67 1−0.763T+67T2 1 - 0.763T + 67T^{2}
71 1+(−10.7+7.77i)T+(21.9−67.5i)T2 1 + (-10.7 + 7.77i)T + (21.9 - 67.5i)T^{2}
73 1+(0.527+1.62i)T+(−59.0+42.9i)T2 1 + (0.527 + 1.62i)T + (-59.0 + 42.9i)T^{2}
79 1+(−1.23−0.898i)T+(24.4+75.1i)T2 1 + (-1.23 - 0.898i)T + (24.4 + 75.1i)T^{2}
83 1+(−12.7+9.23i)T+(25.6−78.9i)T2 1 + (-12.7 + 9.23i)T + (25.6 - 78.9i)T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+(−9.70−7.05i)T+(29.9+92.2i)T2 1 + (-9.70 - 7.05i)T + (29.9 + 92.2i)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

−10.42185018162458223655979563220, −9.077966971646245942884300326976, −8.545151239372250503868021675802, −7.80325616394896683522593331056, −6.44671531535544400556376120123, −6.24132083384146868666677256804, −5.02675774167075249182505265353, −4.06659721110152440853356882766, −3.15257291270504479364362388752, −1.81777555606761575232908817199, 0.77597216124597212794817646144, 2.18135969184387473494269605891, 3.55077765298114147027986070881, 4.43090331516394958823757474460, 4.96068994705655169139085054624, 6.44125587295255857430542522984, 7.00187376665692927817341222094, 8.029873168600945726767336305283, 9.007647835703675106482933507162, 9.850323080719404566839989992066

Graph of the ZZ-function along the critical line