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Properties

Label 2-152-1.1-c3-0-12
Degree 22
Conductor 152152
Sign −1-1
Analytic cond. 8.968298.96829
Root an. cond. 2.994712.99471
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.73·3-s − 18.0·5-s + 0.213·7-s − 4.58·9-s − 3.39·11-s − 90.7·13-s − 85.5·15-s − 2.59·17-s + 19·19-s + 1.01·21-s + 26.6·23-s + 201.·25-s − 149.·27-s + 60.1·29-s − 176.·31-s − 16.0·33-s − 3.85·35-s − 154.·37-s − 429.·39-s + 434.·41-s − 365.·43-s + 82.8·45-s + 204.·47-s − 342.·49-s − 12.2·51-s − 135.·53-s + 61.3·55-s + ⋯
L(s)  = 1  + 0.911·3-s − 1.61·5-s + 0.0115·7-s − 0.169·9-s − 0.0930·11-s − 1.93·13-s − 1.47·15-s − 0.0370·17-s + 0.229·19-s + 0.0104·21-s + 0.241·23-s + 1.61·25-s − 1.06·27-s + 0.384·29-s − 1.02·31-s − 0.0847·33-s − 0.0186·35-s − 0.684·37-s − 1.76·39-s + 1.65·41-s − 1.29·43-s + 0.274·45-s + 0.633·47-s − 0.999·49-s − 0.0337·51-s − 0.351·53-s + 0.150·55-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −1-1
Analytic conductor: 8.968298.96829
Root analytic conductor: 2.994712.99471
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 152, ( :3/2), −1)(2,\ 152,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
19 1−19T 1 - 19T
good3 1−4.73T+27T2 1 - 4.73T + 27T^{2}
5 1+18.0T+125T2 1 + 18.0T + 125T^{2}
7 1−0.213T+343T2 1 - 0.213T + 343T^{2}
11 1+3.39T+1.33e3T2 1 + 3.39T + 1.33e3T^{2}
13 1+90.7T+2.19e3T2 1 + 90.7T + 2.19e3T^{2}
17 1+2.59T+4.91e3T2 1 + 2.59T + 4.91e3T^{2}
23 1−26.6T+1.21e4T2 1 - 26.6T + 1.21e4T^{2}
29 1−60.1T+2.43e4T2 1 - 60.1T + 2.43e4T^{2}
31 1+176.T+2.97e4T2 1 + 176.T + 2.97e4T^{2}
37 1+154.T+5.06e4T2 1 + 154.T + 5.06e4T^{2}
41 1−434.T+6.89e4T2 1 - 434.T + 6.89e4T^{2}
43 1+365.T+7.95e4T2 1 + 365.T + 7.95e4T^{2}
47 1−204.T+1.03e5T2 1 - 204.T + 1.03e5T^{2}
53 1+135.T+1.48e5T2 1 + 135.T + 1.48e5T^{2}
59 1−759.T+2.05e5T2 1 - 759.T + 2.05e5T^{2}
61 1−284.T+2.26e5T2 1 - 284.T + 2.26e5T^{2}
67 1−590.T+3.00e5T2 1 - 590.T + 3.00e5T^{2}
71 1+972.T+3.57e5T2 1 + 972.T + 3.57e5T^{2}
73 1−368.T+3.89e5T2 1 - 368.T + 3.89e5T^{2}
79 1−204.T+4.93e5T2 1 - 204.T + 4.93e5T^{2}
83 1+782.T+5.71e5T2 1 + 782.T + 5.71e5T^{2}
89 1−213.T+7.04e5T2 1 - 213.T + 7.04e5T^{2}
97 1+1.21e3T+9.12e5T2 1 + 1.21e3T + 9.12e5T^{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

−12.02598662687045501501526509585, −11.20933989573261919616816403860, −9.818124951287173822617524905056, −8.712379338474662743522963162737, −7.79599978064815841842736831794, −7.13871577742217804496965110189, −5.05181218860378821487185358045, −3.78771939816173543288843192489, −2.62957810903665442177864465320, 0, 2.62957810903665442177864465320, 3.78771939816173543288843192489, 5.05181218860378821487185358045, 7.13871577742217804496965110189, 7.79599978064815841842736831794, 8.712379338474662743522963162737, 9.818124951287173822617524905056, 11.20933989573261919616816403860, 12.02598662687045501501526509585

Graph of the ZZ-function along the critical line