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Properties

Label 2-33e2-99.85-c0-0-2
Degree 22
Conductor 10891089
Sign −0.216+0.976i-0.216 + 0.976i
Analytic cond. 0.5434810.543481
Root an. cond. 0.7372120.737212
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  + (0.575 − 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.831 − 1.14i)6-s + (−0.294 − 1.38i)7-s + (0.913 + 0.406i)9-s + 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (1.05 − 0.946i)18-s + (0.913 + 0.406i)20-s − 1.41i·21-s + ⋯
L(s)  = 1  + (0.575 − 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.831 − 1.14i)6-s + (−0.294 − 1.38i)7-s + (0.913 + 0.406i)9-s + 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (1.05 − 0.946i)18-s + (0.913 + 0.406i)20-s − 1.41i·21-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.216+0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.216+0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.216+0.976i-0.216 + 0.976i
Analytic conductor: 0.5434810.543481
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1089(481,⋅)\chi_{1089} (481, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :0), −0.216+0.976i)(2,\ 1089,\ (\ :0),\ -0.216 + 0.976i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6614574201.661457420
L(12)L(\frac12) ≈\approx 1.6614574201.661457420
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)
bad3 1+(−0.978−0.207i)T 1 + (-0.978 - 0.207i)T
11 1 1
good2 1+(−0.575+1.29i)T+(−0.669−0.743i)T2 1 + (-0.575 + 1.29i)T + (-0.669 - 0.743i)T^{2}
5 1+(0.913−0.406i)T+(0.669−0.743i)T2 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2}
7 1+(0.294+1.38i)T+(−0.913+0.406i)T2 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2}
13 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+(−0.294−1.38i)T+(−0.913+0.406i)T2 1 + (-0.294 - 1.38i)T + (-0.913 + 0.406i)T^{2}
31 1+(0.104+0.994i)T+(−0.978+0.207i)T2 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2}
37 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
41 1+(−0.913−0.406i)T2 1 + (-0.913 - 0.406i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1+(0.669−0.743i)T+(−0.104−0.994i)T2 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2}
53 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
59 1+(−0.669−0.743i)T+(−0.104+0.994i)T2 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2}
61 1+(1.40+0.147i)T+(0.978+0.207i)T2 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
73 1+(−1.34−0.437i)T+(0.809+0.587i)T2 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2}
79 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
83 1+(1.40+0.147i)T+(0.978+0.207i)T2 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.913−0.406i)T+(0.669+0.743i)T2 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)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

−10.05658802605214049037593533980, −9.358663546949343802614499168077, −8.096725407880222243571027609778, −7.46831683073283949944697811559, −6.76074624274113714801741531640, −4.88394255065497867681319442351, −4.07855154903981425223106410577, −3.54346967428513551941942116057, −2.81815202112658576159413782599, −1.38697578060642748414256129423, 2.10278417985452916650104872269, 3.38439138241093233174617534949, 4.31783817903507970963174537084, 5.22399142127878345734191716243, 6.18975417134652901392379484720, 6.97755755403193247699258928069, 7.914054366546055436269536602337, 8.355140624394309363270090957508, 9.022068543570754253671645558284, 9.954790345350468424118392704126

Graph of the ZZ-function along the critical line