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Properties

Label 2-1152-1.1-c3-0-24
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  − 2.58·5-s + 33.7·7-s + 33.1·11-s + 4.33·13-s − 11.1·17-s + 121.·19-s − 14.8·23-s − 118.·25-s + 272.·29-s + 165.·31-s − 87.1·35-s + 30.5·37-s − 400.·41-s + 274.·43-s − 487.·47-s + 795.·49-s − 208.·53-s − 85.6·55-s − 369.·59-s − 411.·61-s − 11.1·65-s + 407.·67-s − 262.·71-s + 562.·73-s + 1.11e3·77-s − 955.·79-s − 669.·83-s + ⋯
L(s)  = 1  − 0.231·5-s + 1.82·7-s + 0.909·11-s + 0.0924·13-s − 0.159·17-s + 1.47·19-s − 0.134·23-s − 0.946·25-s + 1.74·29-s + 0.956·31-s − 0.421·35-s + 0.135·37-s − 1.52·41-s + 0.974·43-s − 1.51·47-s + 2.32·49-s − 0.540·53-s − 0.210·55-s − 0.816·59-s − 0.864·61-s − 0.0213·65-s + 0.742·67-s − 0.439·71-s + 0.902·73-s + 1.65·77-s − 1.36·79-s − 0.885·83-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 2.9863612172.986361217
L(12)L(\frac12) ≈\approx 2.9863612172.986361217
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+2.58T+125T2 1 + 2.58T + 125T^{2}
7 1−33.7T+343T2 1 - 33.7T + 343T^{2}
11 1−33.1T+1.33e3T2 1 - 33.1T + 1.33e3T^{2}
13 1−4.33T+2.19e3T2 1 - 4.33T + 2.19e3T^{2}
17 1+11.1T+4.91e3T2 1 + 11.1T + 4.91e3T^{2}
19 1−121.T+6.85e3T2 1 - 121.T + 6.85e3T^{2}
23 1+14.8T+1.21e4T2 1 + 14.8T + 1.21e4T^{2}
29 1−272.T+2.43e4T2 1 - 272.T + 2.43e4T^{2}
31 1−165.T+2.97e4T2 1 - 165.T + 2.97e4T^{2}
37 1−30.5T+5.06e4T2 1 - 30.5T + 5.06e4T^{2}
41 1+400.T+6.89e4T2 1 + 400.T + 6.89e4T^{2}
43 1−274.T+7.95e4T2 1 - 274.T + 7.95e4T^{2}
47 1+487.T+1.03e5T2 1 + 487.T + 1.03e5T^{2}
53 1+208.T+1.48e5T2 1 + 208.T + 1.48e5T^{2}
59 1+369.T+2.05e5T2 1 + 369.T + 2.05e5T^{2}
61 1+411.T+2.26e5T2 1 + 411.T + 2.26e5T^{2}
67 1−407.T+3.00e5T2 1 - 407.T + 3.00e5T^{2}
71 1+262.T+3.57e5T2 1 + 262.T + 3.57e5T^{2}
73 1−562.T+3.89e5T2 1 - 562.T + 3.89e5T^{2}
79 1+955.T+4.93e5T2 1 + 955.T + 4.93e5T^{2}
83 1+669.T+5.71e5T2 1 + 669.T + 5.71e5T^{2}
89 1−1.32e3T+7.04e5T2 1 - 1.32e3T + 7.04e5T^{2}
97 1−1.10e3T+9.12e5T2 1 - 1.10e3T + 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

−9.384177408161764268504258348863, −8.385909899373757885425159506058, −7.959739680720633738064353296400, −7.04273856841079049414846522058, −6.02023273732905480751728746594, −4.94415017739941566032992908067, −4.40093238585390962515350028444, −3.20685936785424956192429632618, −1.81141960225062860285732602125, −0.983671779276637201073792316755, 0.983671779276637201073792316755, 1.81141960225062860285732602125, 3.20685936785424956192429632618, 4.40093238585390962515350028444, 4.94415017739941566032992908067, 6.02023273732905480751728746594, 7.04273856841079049414846522058, 7.959739680720633738064353296400, 8.385909899373757885425159506058, 9.384177408161764268504258348863

Graph of the ZZ-function along the critical line