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Properties

Label 2-192-1.1-c7-0-20
Degree 22
Conductor 192192
Sign −1-1
Analytic cond. 59.977959.9779
Root an. cond. 7.744547.74454
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s − 390·5-s − 64·7-s + 729·9-s + 948·11-s + 5.09e3·13-s − 1.05e4·15-s + 2.83e4·17-s + 8.62e3·19-s − 1.72e3·21-s − 1.52e4·23-s + 7.39e4·25-s + 1.96e4·27-s − 3.65e4·29-s − 2.76e5·31-s + 2.55e4·33-s + 2.49e4·35-s − 2.68e5·37-s + 1.37e5·39-s − 6.29e5·41-s − 6.85e5·43-s − 2.84e5·45-s + 5.83e5·47-s − 8.19e5·49-s + 7.66e5·51-s + 4.28e5·53-s − 3.69e5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.39·5-s − 0.0705·7-s + 1/3·9-s + 0.214·11-s + 0.643·13-s − 0.805·15-s + 1.40·17-s + 0.288·19-s − 0.0407·21-s − 0.262·23-s + 0.946·25-s + 0.192·27-s − 0.277·29-s − 1.66·31-s + 0.123·33-s + 0.0984·35-s − 0.871·37-s + 0.371·39-s − 1.42·41-s − 1.31·43-s − 0.465·45-s + 0.819·47-s − 0.995·49-s + 0.809·51-s + 0.394·53-s − 0.299·55-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: −1-1
Analytic conductor: 59.977959.9779
Root analytic conductor: 7.744547.74454
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 192, ( :7/2), −1)(2,\ 192,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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−p3T 1 - p^{3} T
good5 1+78pT+p7T2 1 + 78 p T + p^{7} T^{2}
7 1+64T+p7T2 1 + 64 T + p^{7} T^{2}
11 1−948T+p7T2 1 - 948 T + p^{7} T^{2}
13 1−5098T+p7T2 1 - 5098 T + p^{7} T^{2}
17 1−28386T+p7T2 1 - 28386 T + p^{7} T^{2}
19 1−8620T+p7T2 1 - 8620 T + p^{7} T^{2}
23 1+15288T+p7T2 1 + 15288 T + p^{7} T^{2}
29 1+36510T+p7T2 1 + 36510 T + p^{7} T^{2}
31 1+276808T+p7T2 1 + 276808 T + p^{7} T^{2}
37 1+268526T+p7T2 1 + 268526 T + p^{7} T^{2}
41 1+629718T+p7T2 1 + 629718 T + p^{7} T^{2}
43 1+685772T+p7T2 1 + 685772 T + p^{7} T^{2}
47 1−583296T+p7T2 1 - 583296 T + p^{7} T^{2}
53 1−428058T+p7T2 1 - 428058 T + p^{7} T^{2}
59 1+1306380T+p7T2 1 + 1306380 T + p^{7} T^{2}
61 1+300662T+p7T2 1 + 300662 T + p^{7} T^{2}
67 1−507244T+p7T2 1 - 507244 T + p^{7} T^{2}
71 1−5560632T+p7T2 1 - 5560632 T + p^{7} T^{2}
73 1−1369082T+p7T2 1 - 1369082 T + p^{7} T^{2}
79 1+6913720T+p7T2 1 + 6913720 T + p^{7} T^{2}
83 1−4376748T+p7T2 1 - 4376748 T + p^{7} T^{2}
89 1+8528310T+p7T2 1 + 8528310 T + p^{7} T^{2}
97 1+8826814T+p7T2 1 + 8826814 T + p^{7} 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

−10.84922696098999470447219261631, −9.681961161391134871247961888112, −8.530798418160299560576128144086, −7.82129646082650928019867600119, −6.90271509903548344257030697629, −5.32519708966436334688135072451, −3.85599623626722855000687903434, −3.30650856190990078215543284785, −1.47026575733247437183785310283, 0, 1.47026575733247437183785310283, 3.30650856190990078215543284785, 3.85599623626722855000687903434, 5.32519708966436334688135072451, 6.90271509903548344257030697629, 7.82129646082650928019867600119, 8.530798418160299560576128144086, 9.681961161391134871247961888112, 10.84922696098999470447219261631

Graph of the ZZ-function along the critical line