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Properties

Label 2-1152-1.1-c3-0-29
Degree 22
Conductor 11521152
Sign 11
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.5·5-s + 29.7·7-s + 9.16·11-s − 80.3·13-s + 31.1·17-s + 89.8·19-s + 57.1·23-s + 220.·25-s + 167.·29-s − 270.·31-s + 552.·35-s + 157.·37-s + 404.·41-s − 317.·43-s − 63.1·47-s + 542.·49-s + 616.·53-s + 170.·55-s − 137.·59-s − 200.·61-s − 1.49e3·65-s − 576.·67-s + 305.·71-s − 198.·73-s + 272.·77-s − 335.·79-s − 981.·83-s + ⋯
L(s)  = 1  + 1.66·5-s + 1.60·7-s + 0.251·11-s − 1.71·13-s + 0.444·17-s + 1.08·19-s + 0.518·23-s + 1.76·25-s + 1.06·29-s − 1.56·31-s + 2.66·35-s + 0.699·37-s + 1.53·41-s − 1.12·43-s − 0.196·47-s + 1.58·49-s + 1.59·53-s + 0.417·55-s − 0.304·59-s − 0.420·61-s − 2.84·65-s − 1.05·67-s + 0.510·71-s − 0.319·73-s + 0.403·77-s − 0.477·79-s − 1.29·83-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 11
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 1)(2,\ 1152,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.8087400873.808740087
L(12)L(\frac12) ≈\approx 3.8087400873.808740087
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
3 1 1
good5 1−18.5T+125T2 1 - 18.5T + 125T^{2}
7 1−29.7T+343T2 1 - 29.7T + 343T^{2}
11 1−9.16T+1.33e3T2 1 - 9.16T + 1.33e3T^{2}
13 1+80.3T+2.19e3T2 1 + 80.3T + 2.19e3T^{2}
17 1−31.1T+4.91e3T2 1 - 31.1T + 4.91e3T^{2}
19 1−89.8T+6.85e3T2 1 - 89.8T + 6.85e3T^{2}
23 1−57.1T+1.21e4T2 1 - 57.1T + 1.21e4T^{2}
29 1−167.T+2.43e4T2 1 - 167.T + 2.43e4T^{2}
31 1+270.T+2.97e4T2 1 + 270.T + 2.97e4T^{2}
37 1−157.T+5.06e4T2 1 - 157.T + 5.06e4T^{2}
41 1−404.T+6.89e4T2 1 - 404.T + 6.89e4T^{2}
43 1+317.T+7.95e4T2 1 + 317.T + 7.95e4T^{2}
47 1+63.1T+1.03e5T2 1 + 63.1T + 1.03e5T^{2}
53 1−616.T+1.48e5T2 1 - 616.T + 1.48e5T^{2}
59 1+137.T+2.05e5T2 1 + 137.T + 2.05e5T^{2}
61 1+200.T+2.26e5T2 1 + 200.T + 2.26e5T^{2}
67 1+576.T+3.00e5T2 1 + 576.T + 3.00e5T^{2}
71 1−305.T+3.57e5T2 1 - 305.T + 3.57e5T^{2}
73 1+198.T+3.89e5T2 1 + 198.T + 3.89e5T^{2}
79 1+335.T+4.93e5T2 1 + 335.T + 4.93e5T^{2}
83 1+981.T+5.71e5T2 1 + 981.T + 5.71e5T^{2}
89 1−51.0T+7.04e5T2 1 - 51.0T + 7.04e5T^{2}
97 1−678.T+9.12e5T2 1 - 678.T + 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

−9.522589093918709511695526984132, −8.751891261850584396591259450639, −7.66222641281372018514788489199, −7.05901354685304830183698552476, −5.77827563243256894063278446158, −5.22233497141732088674305687847, −4.54927819706969169864341959802, −2.82284261248844728557922375721, −1.96832593831325954416778781969, −1.10912183581758619118846140617, 1.10912183581758619118846140617, 1.96832593831325954416778781969, 2.82284261248844728557922375721, 4.54927819706969169864341959802, 5.22233497141732088674305687847, 5.77827563243256894063278446158, 7.05901354685304830183698552476, 7.66222641281372018514788489199, 8.751891261850584396591259450639, 9.522589093918709511695526984132

Graph of the ZZ-function along the critical line