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Properties

Label 2-1872-1.1-c3-0-5
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 0.38093269470.3809326947
L(12)L(\frac12) ≈\approx 0.38093269470.3809326947
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

−8.902921733716050199424111499459, −7.958628741693447049171440367269, −7.39157479292450407604174587854, −6.63320909834483675107415026896, −5.63862862452443894890860570511, −4.74149378628280901363420889326, −3.68626123522913615522156940459, −3.22418992306099529151945928712, −1.88231142312022988005247909335, −0.27445774622771076669377282560, 0.27445774622771076669377282560, 1.88231142312022988005247909335, 3.22418992306099529151945928712, 3.68626123522913615522156940459, 4.74149378628280901363420889326, 5.63862862452443894890860570511, 6.63320909834483675107415026896, 7.39157479292450407604174587854, 7.958628741693447049171440367269, 8.902921733716050199424111499459

Graph of the ZZ-function along the critical line