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Properties

Label 2-29988-1.1-c1-0-25
Degree 22
Conductor 2998829988
Sign −1-1
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
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  − 3·5-s − 4·11-s + 13-s + 17-s − 4·23-s + 4·25-s − 29-s − 3·31-s + 2·37-s + 5·41-s + 6·43-s − 9·47-s − 2·53-s + 12·55-s − 59-s + 12·61-s − 3·65-s + 10·67-s − 8·73-s + 5·83-s − 3·85-s + 6·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.20·11-s + 0.277·13-s + 0.242·17-s − 0.834·23-s + 4/5·25-s − 0.185·29-s − 0.538·31-s + 0.328·37-s + 0.780·41-s + 0.914·43-s − 1.31·47-s − 0.274·53-s + 1.61·55-s − 0.130·59-s + 1.53·61-s − 0.372·65-s + 1.22·67-s − 0.936·73-s + 0.548·83-s − 0.325·85-s + 0.635·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: −1-1
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 29988, ( :1/2), −1)(2,\ 29988,\ (\ :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)
bad2 1 1
3 1 1
7 1 1
17 1−T 1 - T
good5 1+3T+pT2 1 + 3 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+T+pT2 1 + T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−5T+pT2 1 - 5 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−15.53332716009287, −14.88420006060256, −14.49292588530748, −13.82923513868280, −13.14521278192261, −12.73717456669143, −12.23416383559746, −11.63324933851931, −11.09652059268236, −10.79801333910899, −9.982758531105922, −9.584123786625779, −8.642325805092381, −8.276690397409191, −7.655681436160770, −7.444836423271714, −6.607625028740991, −5.877540916850418, −5.288524405854922, −4.600241421116700, −3.973224378427199, −3.452783437847950, −2.715187799600650, −1.948981162317962, −0.7888872806357225, 0, 0.7888872806357225, 1.948981162317962, 2.715187799600650, 3.452783437847950, 3.973224378427199, 4.600241421116700, 5.288524405854922, 5.877540916850418, 6.607625028740991, 7.444836423271714, 7.655681436160770, 8.276690397409191, 8.642325805092381, 9.584123786625779, 9.982758531105922, 10.79801333910899, 11.09652059268236, 11.63324933851931, 12.23416383559746, 12.73717456669143, 13.14521278192261, 13.82923513868280, 14.49292588530748, 14.88420006060256, 15.53332716009287

Graph of the ZZ-function along the critical line