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Properties

Label 2-1232-1.1-c3-0-43
Degree 22
Conductor 12321232
Sign 11
Analytic cond. 72.690372.6903
Root an. cond. 8.525868.52586
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  + 9.65·3-s − 3.08·5-s − 7·7-s + 66.1·9-s − 11·11-s + 13.5·13-s − 29.8·15-s − 87.3·17-s + 23.4·19-s − 67.5·21-s + 193.·23-s − 115.·25-s + 378.·27-s + 224.·29-s + 84.0·31-s − 106.·33-s + 21.6·35-s + 138.·37-s + 130.·39-s + 404.·41-s + 138.·43-s − 204.·45-s + 198.·47-s + 49·49-s − 843.·51-s + 423.·53-s + 33.9·55-s + ⋯
L(s)  = 1  + 1.85·3-s − 0.276·5-s − 0.377·7-s + 2.45·9-s − 0.301·11-s + 0.288·13-s − 0.513·15-s − 1.24·17-s + 0.283·19-s − 0.702·21-s + 1.75·23-s − 0.923·25-s + 2.69·27-s + 1.43·29-s + 0.486·31-s − 0.560·33-s + 0.104·35-s + 0.615·37-s + 0.535·39-s + 1.54·41-s + 0.489·43-s − 0.676·45-s + 0.614·47-s + 0.142·49-s − 2.31·51-s + 1.09·53-s + 0.0832·55-s + ⋯

Functional equation

Λ(s)=(1232s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1232s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12321232    =    24⋅7⋅112^{4} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 72.690372.6903
Root analytic conductor: 8.525868.52586
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1232, ( :3/2), 1)(2,\ 1232,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.2652907304.265290730
L(12)L(\frac12) ≈\approx 4.2652907304.265290730
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
7 1+7T 1 + 7T
11 1+11T 1 + 11T
good3 1−9.65T+27T2 1 - 9.65T + 27T^{2}
5 1+3.08T+125T2 1 + 3.08T + 125T^{2}
13 1−13.5T+2.19e3T2 1 - 13.5T + 2.19e3T^{2}
17 1+87.3T+4.91e3T2 1 + 87.3T + 4.91e3T^{2}
19 1−23.4T+6.85e3T2 1 - 23.4T + 6.85e3T^{2}
23 1−193.T+1.21e4T2 1 - 193.T + 1.21e4T^{2}
29 1−224.T+2.43e4T2 1 - 224.T + 2.43e4T^{2}
31 1−84.0T+2.97e4T2 1 - 84.0T + 2.97e4T^{2}
37 1−138.T+5.06e4T2 1 - 138.T + 5.06e4T^{2}
41 1−404.T+6.89e4T2 1 - 404.T + 6.89e4T^{2}
43 1−138.T+7.95e4T2 1 - 138.T + 7.95e4T^{2}
47 1−198.T+1.03e5T2 1 - 198.T + 1.03e5T^{2}
53 1−423.T+1.48e5T2 1 - 423.T + 1.48e5T^{2}
59 1−311.T+2.05e5T2 1 - 311.T + 2.05e5T^{2}
61 1+262.T+2.26e5T2 1 + 262.T + 2.26e5T^{2}
67 1+441.T+3.00e5T2 1 + 441.T + 3.00e5T^{2}
71 1+731.T+3.57e5T2 1 + 731.T + 3.57e5T^{2}
73 1+723.T+3.89e5T2 1 + 723.T + 3.89e5T^{2}
79 1−647.T+4.93e5T2 1 - 647.T + 4.93e5T^{2}
83 1+72.9T+5.71e5T2 1 + 72.9T + 5.71e5T^{2}
89 1+243.T+7.04e5T2 1 + 243.T + 7.04e5T^{2}
97 1−1.61e3T+9.12e5T2 1 - 1.61e3T + 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

−8.938740440866195368360131430843, −8.829314113811933063775135387486, −7.73850448872567094944555260114, −7.21534787953682923233056128026, −6.23184136553215791047383981832, −4.69162685903982842033479751080, −3.97899842052922618543110053169, −2.95233578187767413045587384577, −2.39658078810104901329302519693, −0.998819271263063248124085349073, 0.998819271263063248124085349073, 2.39658078810104901329302519693, 2.95233578187767413045587384577, 3.97899842052922618543110053169, 4.69162685903982842033479751080, 6.23184136553215791047383981832, 7.21534787953682923233056128026, 7.73850448872567094944555260114, 8.829314113811933063775135387486, 8.938740440866195368360131430843

Graph of the ZZ-function along the critical line