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Properties

Label 4-800e2-1.1-c1e2-0-21
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 + 2·7-s + 2·9-s + 8·13-s − 8·17-s − 8·19-s + 4·21-s + 10·23-s + 6·27-s + 16·39-s − 8·41-s + 14·43-s + 6·47-s + 2·49-s − 16·51-s + 8·53-s − 16·57-s − 8·59-s − 16·61-s + 4·63-s − 6·67-s + 20·69-s + 8·73-s + 16·79-s + 11·81-s − 10·83-s + 16·91-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 2/3·9-s + 2.21·13-s − 1.94·17-s − 1.83·19-s + 0.872·21-s + 2.08·23-s + 1.15·27-s + 2.56·39-s − 1.24·41-s + 2.13·43-s + 0.875·47-s + 2/7·49-s − 2.24·51-s + 1.09·53-s − 2.11·57-s − 1.04·59-s − 2.04·61-s + 0.503·63-s − 0.733·67-s + 2.40·69-s + 0.936·73-s + 1.80·79-s + 11/9·81-s − 1.09·83-s + 1.67·91-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 3.6481602883.648160288
L(12)L(\frac12) ≈\approx 3.6481602883.648160288
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 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
37C22C_2^2 1+p2T4 1 + p^{2} T^{4}
41C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
43C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 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}
53C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
67C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+114T2+p2T4 1 + 114 T^{2} + p^{2} T^{4}
73C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 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 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
97C22C_2^2 1−24T+288T2−24pT3+p2T4 1 - 24 T + 288 T^{2} - 24 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

−10.55374786445866496768684134962, −10.29097544765293290287876127753, −9.089996388508825785562720946344, −9.052485826671851602418182376027, −8.747004119777754145033551374780, −8.743942219416152535117199098085, −7.83665134145972727486867128390, −7.83028897057092040245201781733, −6.90187234889065872877206094653, −6.68666952001963490752195364968, −6.20712555634431725100270999448, −5.75823904353993206637646944286, −4.78108752864557783268703417048, −4.65076702289541684965087046332, −4.03416747726069029576543962573, −3.59024636686226889727514752069, −2.89515212615180721093441613575, −2.35336550025024280112408643865, −1.75930866726089681030256218148, −0.959948261472095265651885227825, 0.959948261472095265651885227825, 1.75930866726089681030256218148, 2.35336550025024280112408643865, 2.89515212615180721093441613575, 3.59024636686226889727514752069, 4.03416747726069029576543962573, 4.65076702289541684965087046332, 4.78108752864557783268703417048, 5.75823904353993206637646944286, 6.20712555634431725100270999448, 6.68666952001963490752195364968, 6.90187234889065872877206094653, 7.83028897057092040245201781733, 7.83665134145972727486867128390, 8.743942219416152535117199098085, 8.747004119777754145033551374780, 9.052485826671851602418182376027, 9.089996388508825785562720946344, 10.29097544765293290287876127753, 10.55374786445866496768684134962

Graph of the ZZ-function along the critical line