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Properties

Label 4-86400-1.1-c1e2-0-33
Degree 44
Conductor 8640086400
Sign −1-1
Analytic cond. 5.508935.50893
Root an. cond. 1.532021.53202
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 12·13-s − 16·23-s + 25-s + 27-s − 12·37-s − 12·39-s + 16·47-s + 2·49-s + 12·61-s − 16·69-s − 28·73-s + 75-s + 81-s + 24·83-s + 4·97-s + 8·107-s − 20·109-s − 12·111-s − 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 16·141-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 3.32·13-s − 3.33·23-s + 1/5·25-s + 0.192·27-s − 1.97·37-s − 1.92·39-s + 2.33·47-s + 2/7·49-s + 1.53·61-s − 1.92·69-s − 3.27·73-s + 0.115·75-s + 1/9·81-s + 2.63·83-s + 0.406·97-s + 0.773·107-s − 1.91·109-s − 1.13·111-s − 1.10·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.34·141-s + ⋯

Functional equation

Λ(s)=(86400s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(86400s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8640086400    =    27⋅33⋅522^{7} \cdot 3^{3} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 5.508935.50893
Root analytic conductor: 1.532021.53202
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 86400, ( :1/2,1/2), −1)(4,\ 86400,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 1−T 1 - T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.345822684053937464266140209197, −9.138788409882082860623313179234, −8.312681871958482675893970014801, −7.929044973464791963060046214031, −7.34594365176835478558310775458, −7.18880346144848821197627995561, −6.46041242786302870623045801913, −5.61049576810701540702833296151, −5.28897888504504330389281711769, −4.43140184316888462520939708171, −4.13728859524865050162716547657, −3.20808194979498891977471116338, −2.19282779410354418054159546466, −2.19180563382997996757520036314, 0, 2.19180563382997996757520036314, 2.19282779410354418054159546466, 3.20808194979498891977471116338, 4.13728859524865050162716547657, 4.43140184316888462520939708171, 5.28897888504504330389281711769, 5.61049576810701540702833296151, 6.46041242786302870623045801913, 7.18880346144848821197627995561, 7.34594365176835478558310775458, 7.929044973464791963060046214031, 8.312681871958482675893970014801, 9.138788409882082860623313179234, 9.345822684053937464266140209197

Graph of the ZZ-function along the critical line