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Properties

Label 4-800e2-1.1-c1e2-0-0
Degree 44
Conductor 640000640000
Sign 11
Analytic cond. 40.806940.8069
Root an. cond. 2.527452.52745
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 6·7-s + 2·9-s − 6·13-s − 2·17-s − 8·19-s − 12·21-s − 2·23-s + 6·27-s + 2·37-s − 12·39-s − 20·41-s + 10·43-s − 6·47-s + 18·49-s − 4·51-s + 10·53-s − 16·57-s + 24·59-s + 4·61-s − 12·63-s − 2·67-s − 4·69-s − 2·73-s − 16·79-s + 11·81-s + 10·83-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.26·7-s + 2/3·9-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 2.61·21-s − 0.417·23-s + 1.15·27-s + 0.328·37-s − 1.92·39-s − 3.12·41-s + 1.52·43-s − 0.875·47-s + 18/7·49-s − 0.560·51-s + 1.37·53-s − 2.11·57-s + 3.12·59-s + 0.512·61-s − 1.51·63-s − 0.244·67-s − 0.481·69-s − 0.234·73-s − 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯

Functional equation

Λ(s)=(640000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(640000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 640000640000    =    210⋅542^{10} \cdot 5^{4}
Sign: 11
Analytic conductor: 40.806940.8069
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 640000, ( :1/2,1/2), 1)(4,\ 640000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.99477703550.9947770355
L(12)L(\frac12) ≈\approx 0.99477703550.9947770355
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
5 1 1
good3C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
13C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C2C_2 (1−14T+pT2)(1+4T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−138T2+p2T4 1 - 138 T^{2} + p^{2} T^{4}
73C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
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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

−10.19919946969799408617834836787, −10.12019770827379467208749869707, −9.724690470722904152278049104281, −8.991661113158697161383666627446, −8.987052401958675011738244603072, −8.452724393400733958631181553980, −8.069463989498949098821476824004, −7.38569260731394457793209783807, −6.92574740119835516732910322817, −6.63422051205586832692203262238, −6.44874293123435635520929336145, −5.60819060416649640444834770186, −5.17525279931901536021265829349, −4.24383002231607724826439919838, −4.18371058760003549663218260902, −3.26785303674686534517367493068, −3.09577955010136171888452269471, −2.22325946034601879952506452022, −2.21567402034006199968334476966, −0.42870044250894246003153245453, 0.42870044250894246003153245453, 2.21567402034006199968334476966, 2.22325946034601879952506452022, 3.09577955010136171888452269471, 3.26785303674686534517367493068, 4.18371058760003549663218260902, 4.24383002231607724826439919838, 5.17525279931901536021265829349, 5.60819060416649640444834770186, 6.44874293123435635520929336145, 6.63422051205586832692203262238, 6.92574740119835516732910322817, 7.38569260731394457793209783807, 8.069463989498949098821476824004, 8.452724393400733958631181553980, 8.987052401958675011738244603072, 8.991661113158697161383666627446, 9.724690470722904152278049104281, 10.12019770827379467208749869707, 10.19919946969799408617834836787

Graph of the ZZ-function along the critical line