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Properties

Label 2-43560-1.1-c1-0-44
Degree 22
Conductor 4356043560
Sign −1-1
Analytic cond. 347.828347.828
Root an. cond. 18.650118.6501
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 13-s − 17-s − 3·19-s + 6·23-s + 25-s − 3·29-s − 35-s − 5·37-s − 12·41-s + 8·43-s + 12·47-s − 6·49-s + 2·53-s + 65-s + 12·67-s + 71-s − 8·73-s + 8·79-s − 83-s + 85-s + 6·89-s − 91-s + 3·95-s − 8·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.169·35-s − 0.821·37-s − 1.87·41-s + 1.21·43-s + 1.75·47-s − 6/7·49-s + 0.274·53-s + 0.124·65-s + 1.46·67-s + 0.118·71-s − 0.936·73-s + 0.900·79-s − 0.109·83-s + 0.108·85-s + 0.635·89-s − 0.104·91-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯

Functional equation

Λ(s)=(43560s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43560s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4356043560    =    23⋅32⋅5⋅1122^{3} \cdot 3^{2} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 347.828347.828
Root analytic conductor: 18.650118.6501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 43560, ( :1/2), −1)(2,\ 43560,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1+T 1 + T
11 1 1
good7 1−T+pT2 1 - T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1+3T+pT2 1 + 3 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+5T+pT2 1 + 5 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 1−12T+pT2 1 - 12 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−T+pT2 1 - T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+T+pT2 1 + T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−14.98151387422909, −14.48883772443090, −13.96733380107886, −13.31271189977031, −12.93228237095858, −12.24404885399055, −11.94101650495782, −11.21408570264787, −10.82403783318872, −10.40948947484456, −9.634902166086961, −9.073050529806157, −8.600252321406210, −8.088137124105686, −7.428037022743951, −6.944242111397420, −6.472140779566139, −5.568280558805606, −5.142858454559912, −4.494236545458802, −3.905533613264600, −3.257280121727601, −2.497786084378365, −1.820153631398018, −0.9370139331521517, 0, 0.9370139331521517, 1.820153631398018, 2.497786084378365, 3.257280121727601, 3.905533613264600, 4.494236545458802, 5.142858454559912, 5.568280558805606, 6.472140779566139, 6.944242111397420, 7.428037022743951, 8.088137124105686, 8.600252321406210, 9.073050529806157, 9.634902166086961, 10.40948947484456, 10.82403783318872, 11.21408570264787, 11.94101650495782, 12.24404885399055, 12.93228237095858, 13.31271189977031, 13.96733380107886, 14.48883772443090, 14.98151387422909

Graph of the ZZ-function along the critical line