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Properties

Label 2-1872-1.1-c1-0-15
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 7-s + 6·11-s + 13-s + 3·17-s − 2·19-s + 4·25-s − 6·29-s + 4·31-s + 3·35-s − 7·37-s + 43-s + 3·47-s − 6·49-s + 18·55-s − 6·59-s + 8·61-s + 3·65-s − 14·67-s − 3·71-s + 2·73-s + 6·77-s − 8·79-s + 12·83-s + 9·85-s + 6·89-s + 91-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s + 1.80·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 4/5·25-s − 1.11·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 0.152·43-s + 0.437·47-s − 6/7·49-s + 2.42·55-s − 0.781·59-s + 1.02·61-s + 0.372·65-s − 1.71·67-s − 0.356·71-s + 0.234·73-s + 0.683·77-s − 0.900·79-s + 1.31·83-s + 0.976·85-s + 0.635·89-s + 0.104·91-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 1)(2,\ 1872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6790148652.679014865
L(12)L(\frac12) ≈\approx 2.6790148652.679014865
L(32)L(\frac{3}{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−T 1 - T
good5 1−3T+pT2 1 - 3 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+14T+pT2 1 + 14 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−9.197451332644930457875280868446, −8.727543231328509862646305089705, −7.62631082392611378394973504804, −6.63805072209796513425044648809, −6.10177969443293897115150871480, −5.33155182765537854352160831380, −4.28669590144044996642748248504, −3.33995220257029308986401899415, −1.98547611328597681321826282171, −1.28946814831643962415194675902, 1.28946814831643962415194675902, 1.98547611328597681321826282171, 3.33995220257029308986401899415, 4.28669590144044996642748248504, 5.33155182765537854352160831380, 6.10177969443293897115150871480, 6.63805072209796513425044648809, 7.62631082392611378394973504804, 8.727543231328509862646305089705, 9.197451332644930457875280868446

Graph of the ZZ-function along the critical line