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Properties

Label 2-1872-1.1-c1-0-11
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  + 2·5-s + 2·7-s − 4·11-s − 13-s + 6·19-s + 4·23-s − 25-s + 8·29-s + 2·31-s + 4·35-s + 6·37-s − 6·41-s + 8·43-s + 8·47-s − 3·49-s − 12·53-s − 8·55-s + 4·59-s + 10·61-s − 2·65-s + 2·67-s − 16·71-s + 14·73-s − 8·77-s + 4·79-s − 12·83-s + 6·89-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.37·19-s + 0.834·23-s − 1/5·25-s + 1.48·29-s + 0.359·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s − 1.07·55-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 1.89·71-s + 1.63·73-s − 0.911·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-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 2.2156960422.215696042
L(12)L(\frac12) ≈\approx 2.2156960422.215696042
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 1−2T+pT2 1 - 2 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−4T+pT2 1 - 4 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

−9.374800673600871396369360815682, −8.345885893689355266865433880677, −7.74982624300516502645234312538, −6.91437440844756442579529153190, −5.85428751909866229664883723997, −5.21397578582522565243253635444, −4.55774581547134998480701833621, −3.06816752901810724197585330945, −2.30205596173135729346442665808, −1.06736363009885798558856425967, 1.06736363009885798558856425967, 2.30205596173135729346442665808, 3.06816752901810724197585330945, 4.55774581547134998480701833621, 5.21397578582522565243253635444, 5.85428751909866229664883723997, 6.91437440844756442579529153190, 7.74982624300516502645234312538, 8.345885893689355266865433880677, 9.374800673600871396369360815682

Graph of the ZZ-function along the critical line