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Properties

Label 2-33e2-1.1-c1-0-17
Degree 22
Conductor 10891089
Sign −1-1
Analytic cond. 8.695708.69570
Root an. cond. 2.948842.94884
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 4·5-s − 7-s + 8·10-s + 2·13-s + 2·14-s − 4·16-s + 4·17-s + 3·19-s − 8·20-s − 2·23-s + 11·25-s − 4·26-s − 2·28-s + 6·29-s − 5·31-s + 8·32-s − 8·34-s + 4·35-s + 3·37-s − 6·38-s − 2·41-s − 12·43-s + 4·46-s − 2·47-s − 6·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s − 0.377·7-s + 2.52·10-s + 0.554·13-s + 0.534·14-s − 16-s + 0.970·17-s + 0.688·19-s − 1.78·20-s − 0.417·23-s + 11/5·25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.898·31-s + 1.41·32-s − 1.37·34-s + 0.676·35-s + 0.493·37-s − 0.973·38-s − 0.312·41-s − 1.82·43-s + 0.589·46-s − 0.291·47-s − 6/7·49-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 8.695708.69570
Root analytic conductor: 2.948842.94884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1089, ( :1/2), −1)(2,\ 1089,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
11 1 1
good2 1+pT+pT2 1 + p T + p T^{2}
5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+3T+pT2 1 + 3 T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−5T+pT2 1 - 5 T + p 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

−9.427976724918087578878267878502, −8.345816310960391722598141882781, −8.118926450037046023972998230501, −7.28899889706555778737663428863, −6.56051958738602663681229923169, −5.05774622025620960162902603755, −3.93444342612617848308073041065, −3.09166168739687900450908810991, −1.23603098910894570836676745841, 0, 1.23603098910894570836676745841, 3.09166168739687900450908810991, 3.93444342612617848308073041065, 5.05774622025620960162902603755, 6.56051958738602663681229923169, 7.28899889706555778737663428863, 8.118926450037046023972998230501, 8.345816310960391722598141882781, 9.427976724918087578878267878502

Graph of the ZZ-function along the critical line