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Properties

Label 2-1872-1.1-c1-0-12
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  + 4·5-s − 4·7-s + 4·11-s − 13-s + 8·23-s + 11·25-s − 8·29-s − 4·31-s − 16·35-s + 6·37-s + 12·41-s + 8·43-s + 4·47-s + 9·49-s + 16·55-s − 4·59-s − 2·61-s − 4·65-s + 8·67-s + 4·71-s − 10·73-s − 16·77-s + 4·79-s − 12·83-s − 12·89-s + 4·91-s + 14·97-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s + 1.20·11-s − 0.277·13-s + 1.66·23-s + 11/5·25-s − 1.48·29-s − 0.718·31-s − 2.70·35-s + 0.986·37-s + 1.87·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s + 2.15·55-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.977·67-s + 0.474·71-s − 1.17·73-s − 1.82·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s + 0.419·91-s + 1.42·97-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 2.2497620442.249762044
L(12)L(\frac12) ≈\approx 2.2497620442.249762044
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 1−4T+pT2 1 - 4 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+10T+pT2 1 + 10 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+12T+pT2 1 + 12 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−9.283309900561460264303460080023, −9.043893881767278031288410702249, −7.35411058268733168359690626917, −6.72499661989682939095500288427, −5.99934179950640698199375325057, −5.54340160378585790784255817080, −4.24052892769095243824819538025, −3.14438875263608746657555973525, −2.30677622130328505633184150541, −1.07183735496432547021055620536, 1.07183735496432547021055620536, 2.30677622130328505633184150541, 3.14438875263608746657555973525, 4.24052892769095243824819538025, 5.54340160378585790784255817080, 5.99934179950640698199375325057, 6.72499661989682939095500288427, 7.35411058268733168359690626917, 9.043893881767278031288410702249, 9.283309900561460264303460080023

Graph of the ZZ-function along the critical line