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Properties

Label 2-1872-1.1-c3-0-33
Degree 22
Conductor 18721872
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.3·5-s − 15.3·7-s + 24.9·11-s + 13·13-s + 65.5·17-s + 73.1·19-s + 28.5·23-s + 54.0·25-s + 220.·29-s − 138.·31-s − 205.·35-s − 354.·37-s + 297.·41-s + 9.62·43-s − 219.·47-s − 106.·49-s + 189.·53-s + 333.·55-s + 329.·59-s − 838.·61-s + 173.·65-s + 386.·67-s + 664.·71-s + 248.·73-s − 383.·77-s + 1.26e3·79-s − 157.·83-s + ⋯
L(s)  = 1  + 1.19·5-s − 0.830·7-s + 0.682·11-s + 0.277·13-s + 0.934·17-s + 0.883·19-s + 0.259·23-s + 0.432·25-s + 1.41·29-s − 0.805·31-s − 0.993·35-s − 1.57·37-s + 1.13·41-s + 0.0341·43-s − 0.681·47-s − 0.310·49-s + 0.490·53-s + 0.816·55-s + 0.726·59-s − 1.76·61-s + 0.331·65-s + 0.705·67-s + 1.11·71-s + 0.398·73-s − 0.566·77-s + 1.80·79-s − 0.208·83-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: 11
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: 00
Selberg data: (2, 1872, ( :3/2), 1)(2,\ 1872,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.9683949892.968394989
L(12)L(\frac12) ≈\approx 2.9683949892.968394989
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 1−13.3T+125T2 1 - 13.3T + 125T^{2}
7 1+15.3T+343T2 1 + 15.3T + 343T^{2}
11 1−24.9T+1.33e3T2 1 - 24.9T + 1.33e3T^{2}
17 1−65.5T+4.91e3T2 1 - 65.5T + 4.91e3T^{2}
19 1−73.1T+6.85e3T2 1 - 73.1T + 6.85e3T^{2}
23 1−28.5T+1.21e4T2 1 - 28.5T + 1.21e4T^{2}
29 1−220.T+2.43e4T2 1 - 220.T + 2.43e4T^{2}
31 1+138.T+2.97e4T2 1 + 138.T + 2.97e4T^{2}
37 1+354.T+5.06e4T2 1 + 354.T + 5.06e4T^{2}
41 1−297.T+6.89e4T2 1 - 297.T + 6.89e4T^{2}
43 1−9.62T+7.95e4T2 1 - 9.62T + 7.95e4T^{2}
47 1+219.T+1.03e5T2 1 + 219.T + 1.03e5T^{2}
53 1−189.T+1.48e5T2 1 - 189.T + 1.48e5T^{2}
59 1−329.T+2.05e5T2 1 - 329.T + 2.05e5T^{2}
61 1+838.T+2.26e5T2 1 + 838.T + 2.26e5T^{2}
67 1−386.T+3.00e5T2 1 - 386.T + 3.00e5T^{2}
71 1−664.T+3.57e5T2 1 - 664.T + 3.57e5T^{2}
73 1−248.T+3.89e5T2 1 - 248.T + 3.89e5T^{2}
79 1−1.26e3T+4.93e5T2 1 - 1.26e3T + 4.93e5T^{2}
83 1+157.T+5.71e5T2 1 + 157.T + 5.71e5T^{2}
89 1+774.T+7.04e5T2 1 + 774.T + 7.04e5T^{2}
97 1−1.05e3T+9.12e5T2 1 - 1.05e3T + 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

−9.145802262897093197403581091646, −8.164651389729053288420877211121, −7.10061831436141253845213338292, −6.43037197773822141927867403422, −5.74763535301356381520756427143, −5.00736294209181984914737642352, −3.69366743423158997652602608756, −2.96834246161620011156434193081, −1.79743085641532931659034312279, −0.837844445562178959551874520853, 0.837844445562178959551874520853, 1.79743085641532931659034312279, 2.96834246161620011156434193081, 3.69366743423158997652602608756, 5.00736294209181984914737642352, 5.74763535301356381520756427143, 6.43037197773822141927867403422, 7.10061831436141253845213338292, 8.164651389729053288420877211121, 9.145802262897093197403581091646

Graph of the ZZ-function along the critical line