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Properties

Label 2-33e2-99.79-c0-0-0
Degree 22
Conductor 10891089
Sign −0.460−0.887i-0.460 - 0.887i
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  + (−1.05 + 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (1.34 − 0.437i)6-s + (−0.575 − 1.29i)7-s + (0.669 + 0.743i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (−1.40 − 0.147i)18-s + (0.669 + 0.743i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (−1.05 + 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (1.34 − 0.437i)6-s + (−0.575 − 1.29i)7-s + (0.669 + 0.743i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (−1.40 − 0.147i)18-s + (0.669 + 0.743i)20-s + 1.41i·21-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.460−0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.460−0.887i-0.460 - 0.887i
Analytic conductor: 0.5434810.543481
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1089(475,⋅)\chi_{1089} (475, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :0), −0.460−0.887i)(2,\ 1089,\ (\ :0),\ -0.460 - 0.887i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.26366147310.2636614731
L(12)L(\frac12) ≈\approx 0.26366147310.2636614731
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+(0.913+0.406i)T 1 + (0.913 + 0.406i)T
11 1 1
good2 1+(1.05−0.946i)T+(0.104−0.994i)T2 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2}
5 1+(0.669−0.743i)T+(−0.104−0.994i)T2 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2}
7 1+(0.575+1.29i)T+(−0.669+0.743i)T2 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2}
13 1+(−0.913+0.406i)T2 1 + (-0.913 + 0.406i)T^{2}
17 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
19 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
23 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
29 1+(−0.575−1.29i)T+(−0.669+0.743i)T2 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2}
31 1+(0.978−0.207i)T+(0.913−0.406i)T2 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2}
37 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
41 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(−0.104−0.994i)T+(−0.978+0.207i)T2 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2}
53 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
59 1+(0.104−0.994i)T+(−0.978−0.207i)T2 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2}
61 1+(0.294−1.38i)T+(−0.913−0.406i)T2 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)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+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
73 1+(0.831−1.14i)T+(−0.309−0.951i)T2 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2}
79 1+(0.104−0.994i)T2 1 + (0.104 - 0.994i)T^{2}
83 1+(0.294−1.38i)T+(−0.913−0.406i)T2 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.669−0.743i)T+(−0.104+0.994i)T2 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)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

−10.37633005919439845076043536806, −9.564108643939341932419410131596, −8.432555368519922454831476258533, −7.46841669026750260888104470187, −7.14659949396433971246261042802, −6.59863011744108901469845321904, −5.65641789297218858539804508866, −4.30854802329474250784738729025, −3.25997606377375848841490132699, −1.07513108890654075612351258585, 0.47408370384288265223877146384, 2.09084264478343865385506983788, 3.35859812315356422885650446206, 4.51327595723152721171593496064, 5.55057729335510585057097802644, 6.24400564130135902279841143065, 7.63777735757051500568616941818, 8.548253263465825223866503946233, 9.198835227389620170663571675632, 9.735819451222735331028672342442

Graph of the ZZ-function along the critical line