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Properties

Label 2-1872-1.1-c3-0-70
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
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  − 1.46·5-s + 8.39·7-s + 34.7·11-s − 13·13-s + 108.·17-s − 143.·19-s − 128.·23-s − 122.·25-s + 18.8·29-s + 78.5·31-s − 12.2·35-s − 327.·37-s − 327.·41-s + 336.·43-s + 99.2·47-s − 272.·49-s + 686.·53-s − 50.9·55-s − 242.·59-s − 644.·61-s + 19.0·65-s + 871.·67-s + 100.·71-s + 604.·73-s + 291.·77-s − 1.07e3·79-s + 741.·83-s + ⋯
L(s)  = 1  − 0.130·5-s + 0.453·7-s + 0.953·11-s − 0.277·13-s + 1.54·17-s − 1.72·19-s − 1.16·23-s − 0.982·25-s + 0.120·29-s + 0.455·31-s − 0.0593·35-s − 1.45·37-s − 1.24·41-s + 1.19·43-s + 0.308·47-s − 0.794·49-s + 1.77·53-s − 0.124·55-s − 0.534·59-s − 1.35·61-s + 0.0363·65-s + 1.58·67-s + 0.167·71-s + 0.969·73-s + 0.432·77-s − 1.52·79-s + 0.980·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :3/2), −1)(2,\ 1872,\ (\ :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
13 1+13T 1 + 13T
good5 1+1.46T+125T2 1 + 1.46T + 125T^{2}
7 1−8.39T+343T2 1 - 8.39T + 343T^{2}
11 1−34.7T+1.33e3T2 1 - 34.7T + 1.33e3T^{2}
17 1−108.T+4.91e3T2 1 - 108.T + 4.91e3T^{2}
19 1+143.T+6.85e3T2 1 + 143.T + 6.85e3T^{2}
23 1+128.T+1.21e4T2 1 + 128.T + 1.21e4T^{2}
29 1−18.8T+2.43e4T2 1 - 18.8T + 2.43e4T^{2}
31 1−78.5T+2.97e4T2 1 - 78.5T + 2.97e4T^{2}
37 1+327.T+5.06e4T2 1 + 327.T + 5.06e4T^{2}
41 1+327.T+6.89e4T2 1 + 327.T + 6.89e4T^{2}
43 1−336.T+7.95e4T2 1 - 336.T + 7.95e4T^{2}
47 1−99.2T+1.03e5T2 1 - 99.2T + 1.03e5T^{2}
53 1−686.T+1.48e5T2 1 - 686.T + 1.48e5T^{2}
59 1+242.T+2.05e5T2 1 + 242.T + 2.05e5T^{2}
61 1+644.T+2.26e5T2 1 + 644.T + 2.26e5T^{2}
67 1−871.T+3.00e5T2 1 - 871.T + 3.00e5T^{2}
71 1−100.T+3.57e5T2 1 - 100.T + 3.57e5T^{2}
73 1−604.T+3.89e5T2 1 - 604.T + 3.89e5T^{2}
79 1+1.07e3T+4.93e5T2 1 + 1.07e3T + 4.93e5T^{2}
83 1−741.T+5.71e5T2 1 - 741.T + 5.71e5T^{2}
89 1−501.T+7.04e5T2 1 - 501.T + 7.04e5T^{2}
97 1+1.56e3T+9.12e5T2 1 + 1.56e3T + 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

−8.360086228596442890697524805437, −7.85336744706432291335002852031, −6.86514895117093578021830889653, −6.10307184766743378180356063655, −5.26864437116058283106768035677, −4.20642827090801170623272426712, −3.62038399299676803120515399977, −2.26083533964262791808071566392, −1.36031215057164922009612106105, 0, 1.36031215057164922009612106105, 2.26083533964262791808071566392, 3.62038399299676803120515399977, 4.20642827090801170623272426712, 5.26864437116058283106768035677, 6.10307184766743378180356063655, 6.86514895117093578021830889653, 7.85336744706432291335002852031, 8.360086228596442890697524805437

Graph of the ZZ-function along the critical line