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Properties

Label 4-1280e2-1.1-c1e2-0-17
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
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 − 4·5-s − 2·7-s + 2·9-s + 6·13-s − 8·15-s − 6·17-s + 12·19-s − 4·21-s − 6·23-s + 11·25-s + 6·27-s + 8·35-s + 6·37-s + 12·39-s − 12·41-s + 6·43-s − 8·45-s + 18·47-s + 2·49-s − 12·51-s + 10·53-s + 24·57-s + 20·59-s − 24·61-s − 4·63-s − 24·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.755·7-s + 2/3·9-s + 1.66·13-s − 2.06·15-s − 1.45·17-s + 2.75·19-s − 0.872·21-s − 1.25·23-s + 11/5·25-s + 1.15·27-s + 1.35·35-s + 0.986·37-s + 1.92·39-s − 1.87·41-s + 0.914·43-s − 1.19·45-s + 2.62·47-s + 2/7·49-s − 1.68·51-s + 1.37·53-s + 3.17·57-s + 2.60·59-s − 3.07·61-s − 0.503·63-s − 2.97·65-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2521343342.252134334
L(12)L(\frac12) ≈\approx 2.2521343342.252134334
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
5C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
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 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
23C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 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 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
61C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
67C22C_2^2 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)(1+16T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 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

−9.539994810994042521246872505391, −9.520284011433220289967643089439, −8.873495929082855836628155208453, −8.617914218246992681510748451867, −8.319963941414979635705062489178, −7.954351199074866528575922904573, −7.43613424880558753341772018411, −7.16343907583786005929418723647, −6.76078023017890266240201548927, −6.32522873187797143820731578818, −5.53429960314262907686507309015, −5.38203084575206009875689790453, −4.31392515244849746139924828302, −4.16015594184457010061608941458, −3.81078526827458449596811091803, −3.25006326354692888511777464048, −2.93550550580576916165475114295, −2.39747503569600525383894734055, −1.30235978816778227887700498647, −0.65582297781718012055007100399, 0.65582297781718012055007100399, 1.30235978816778227887700498647, 2.39747503569600525383894734055, 2.93550550580576916165475114295, 3.25006326354692888511777464048, 3.81078526827458449596811091803, 4.16015594184457010061608941458, 4.31392515244849746139924828302, 5.38203084575206009875689790453, 5.53429960314262907686507309015, 6.32522873187797143820731578818, 6.76078023017890266240201548927, 7.16343907583786005929418723647, 7.43613424880558753341772018411, 7.954351199074866528575922904573, 8.319963941414979635705062489178, 8.617914218246992681510748451867, 8.873495929082855836628155208453, 9.520284011433220289967643089439, 9.539994810994042521246872505391

Graph of the ZZ-function along the critical line