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Properties

Label 2-1152-1.1-c3-0-38
Degree 22
Conductor 11521152
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.64·5-s − 4.98·7-s − 31.2·11-s + 46.5·13-s + 29.2·17-s + 87.1·19-s − 38.6·23-s − 50.2·25-s + 131.·29-s + 137.·31-s + 43.0·35-s + 51.9·37-s − 226.·41-s − 16.7·43-s + 110.·47-s − 318.·49-s + 142.·53-s + 270.·55-s − 547.·59-s + 9.17·61-s − 402.·65-s − 22.1·67-s − 1.16e3·71-s + 317.·73-s + 155.·77-s − 958.·79-s − 207.·83-s + ⋯
L(s)  = 1  − 0.773·5-s − 0.269·7-s − 0.856·11-s + 0.993·13-s + 0.416·17-s + 1.05·19-s − 0.350·23-s − 0.401·25-s + 0.839·29-s + 0.795·31-s + 0.208·35-s + 0.230·37-s − 0.862·41-s − 0.0593·43-s + 0.341·47-s − 0.927·49-s + 0.368·53-s + 0.662·55-s − 1.20·59-s + 0.0192·61-s − 0.768·65-s − 0.0403·67-s − 1.94·71-s + 0.508·73-s + 0.230·77-s − 1.36·79-s − 0.273·83-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: −1-1
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: 11
Selberg data: (2, 1152, ( :3/2), −1)(2,\ 1152,\ (\ :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
3 1 1
good5 1+8.64T+125T2 1 + 8.64T + 125T^{2}
7 1+4.98T+343T2 1 + 4.98T + 343T^{2}
11 1+31.2T+1.33e3T2 1 + 31.2T + 1.33e3T^{2}
13 1−46.5T+2.19e3T2 1 - 46.5T + 2.19e3T^{2}
17 1−29.2T+4.91e3T2 1 - 29.2T + 4.91e3T^{2}
19 1−87.1T+6.85e3T2 1 - 87.1T + 6.85e3T^{2}
23 1+38.6T+1.21e4T2 1 + 38.6T + 1.21e4T^{2}
29 1−131.T+2.43e4T2 1 - 131.T + 2.43e4T^{2}
31 1−137.T+2.97e4T2 1 - 137.T + 2.97e4T^{2}
37 1−51.9T+5.06e4T2 1 - 51.9T + 5.06e4T^{2}
41 1+226.T+6.89e4T2 1 + 226.T + 6.89e4T^{2}
43 1+16.7T+7.95e4T2 1 + 16.7T + 7.95e4T^{2}
47 1−110.T+1.03e5T2 1 - 110.T + 1.03e5T^{2}
53 1−142.T+1.48e5T2 1 - 142.T + 1.48e5T^{2}
59 1+547.T+2.05e5T2 1 + 547.T + 2.05e5T^{2}
61 1−9.17T+2.26e5T2 1 - 9.17T + 2.26e5T^{2}
67 1+22.1T+3.00e5T2 1 + 22.1T + 3.00e5T^{2}
71 1+1.16e3T+3.57e5T2 1 + 1.16e3T + 3.57e5T^{2}
73 1−317.T+3.89e5T2 1 - 317.T + 3.89e5T^{2}
79 1+958.T+4.93e5T2 1 + 958.T + 4.93e5T^{2}
83 1+207.T+5.71e5T2 1 + 207.T + 5.71e5T^{2}
89 1+97.3T+7.04e5T2 1 + 97.3T + 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

−8.908947255209451755568901722367, −8.066330387742725544778552939993, −7.56649638373872485199248230935, −6.48589540956258302582289408187, −5.62407828629723395306772611233, −4.62660159865972696539602193531, −3.61007776157383569494315271122, −2.82696078355623616342496341354, −1.26366655880377889535661353911, 0, 1.26366655880377889535661353911, 2.82696078355623616342496341354, 3.61007776157383569494315271122, 4.62660159865972696539602193531, 5.62407828629723395306772611233, 6.48589540956258302582289408187, 7.56649638373872485199248230935, 8.066330387742725544778552939993, 8.908947255209451755568901722367

Graph of the ZZ-function along the critical line