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Properties

Label 4-640e2-1.1-c1e2-0-15
Degree 44
Conductor 409600409600
Sign 11
Analytic cond. 26.116426.1164
Root an. cond. 2.260622.26062
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  − 4·3-s + 4·5-s + 6·7-s + 6·9-s − 2·11-s − 16·15-s + 2·17-s − 6·19-s − 24·21-s + 2·23-s + 11·25-s + 4·27-s + 14·29-s + 8·33-s + 24·35-s + 24·45-s + 14·47-s + 18·49-s − 8·51-s + 16·53-s − 8·55-s + 24·57-s + 6·59-s + 2·61-s + 36·63-s − 8·69-s + 6·73-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s + 2.26·7-s + 2·9-s − 0.603·11-s − 4.13·15-s + 0.485·17-s − 1.37·19-s − 5.23·21-s + 0.417·23-s + 11/5·25-s + 0.769·27-s + 2.59·29-s + 1.39·33-s + 4.05·35-s + 3.57·45-s + 2.04·47-s + 18/7·49-s − 1.12·51-s + 2.19·53-s − 1.07·55-s + 3.17·57-s + 0.781·59-s + 0.256·61-s + 4.53·63-s − 0.963·69-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=(409600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(409600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 409600409600    =    214⋅522^{14} \cdot 5^{2}
Sign: 11
Analytic conductor: 26.116426.1164
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 409600, ( :1/2,1/2), 1)(4,\ 409600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5757882681.575788268
L(12)L(\frac12) ≈\approx 1.5757882681.575788268
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}
good3C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
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+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + 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}
19C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1−10T+pT2)(1−4T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} )
31C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
37C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
41C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
53C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
59C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−12T+pT2)(1+10T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1+22T+242T2+22pT3+p2T4 1 + 22 T + 242 T^{2} + 22 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

−10.71939659000990607735654071823, −10.58492815345916074848767081043, −10.11564257766476204471775399216, −9.956232091402147762700204742524, −8.789003204104276890135357857383, −8.675000093834449598913181619331, −8.384707588473095304354851238452, −7.64372194904857167935923136168, −6.84697119957715395753941088827, −6.78124025164253148360976191081, −5.96366063611134929173051975689, −5.81136509128689591171595016118, −5.35469621317564831575329942606, −5.05562347420124196102882738936, −4.64405349424528611062344707573, −4.21409684340139055725408044568, −2.60426345556077271494856877740, −2.42843520072922074682998445031, −1.35578562905217315664829285088, −0.915607802733132282268634140246, 0.915607802733132282268634140246, 1.35578562905217315664829285088, 2.42843520072922074682998445031, 2.60426345556077271494856877740, 4.21409684340139055725408044568, 4.64405349424528611062344707573, 5.05562347420124196102882738936, 5.35469621317564831575329942606, 5.81136509128689591171595016118, 5.96366063611134929173051975689, 6.78124025164253148360976191081, 6.84697119957715395753941088827, 7.64372194904857167935923136168, 8.384707588473095304354851238452, 8.675000093834449598913181619331, 8.789003204104276890135357857383, 9.956232091402147762700204742524, 10.11564257766476204471775399216, 10.58492815345916074848767081043, 10.71939659000990607735654071823

Graph of the ZZ-function along the critical line