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Properties

Label 2-1872-1.1-c1-0-5
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  − 2·7-s + 13-s + 6·17-s − 2·19-s − 5·25-s + 6·29-s − 2·31-s + 2·37-s + 12·41-s + 4·43-s − 3·49-s − 6·53-s + 12·59-s + 2·61-s + 10·67-s + 12·71-s + 14·73-s − 8·79-s + 12·83-s − 2·91-s − 10·97-s − 18·101-s + 16·103-s − 12·107-s + 14·109-s − 6·113-s − 12·119-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.277·13-s + 1.45·17-s − 0.458·19-s − 25-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.900·79-s + 1.31·83-s − 0.209·91-s − 1.01·97-s − 1.79·101-s + 1.57·103-s − 1.16·107-s + 1.34·109-s − 0.564·113-s − 1.10·119-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 1.5890406761.589040676
L(12)L(\frac12) ≈\approx 1.5890406761.589040676
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+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 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+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−14T+pT2 1 - 14 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+pT2 1 + 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

−9.485148325596294183816572559050, −8.330201821850417900078578930922, −7.76963269770720756784752102354, −6.78013921434981904340142936703, −6.05756828657732859884712721948, −5.31787425835571416225740279689, −4.13529224817546208100485895471, −3.37281563192054858635234459773, −2.33566034645582102228367652701, −0.861350648696305200147674327945, 0.861350648696305200147674327945, 2.33566034645582102228367652701, 3.37281563192054858635234459773, 4.13529224817546208100485895471, 5.31787425835571416225740279689, 6.05756828657732859884712721948, 6.78013921434981904340142936703, 7.76963269770720756784752102354, 8.330201821850417900078578930922, 9.485148325596294183816572559050

Graph of the ZZ-function along the critical line