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Properties

Label 4-140e2-1.1-c1e2-0-7
Degree 44
Conductor 1960019600
Sign 11
Analytic cond. 1.249711.24971
Root an. cond. 1.057311.05731
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s + 7-s + 3·9-s + 2·11-s − 12·13-s + 3·15-s − 2·17-s + 3·21-s + 9·23-s + 6·29-s − 2·31-s + 6·33-s + 35-s − 8·37-s − 36·39-s + 10·41-s + 2·43-s + 3·45-s − 8·47-s − 6·49-s − 6·51-s − 4·53-s + 2·55-s + 8·59-s − 7·61-s + 3·63-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 9-s + 0.603·11-s − 3.32·13-s + 0.774·15-s − 0.485·17-s + 0.654·21-s + 1.87·23-s + 1.11·29-s − 0.359·31-s + 1.04·33-s + 0.169·35-s − 1.31·37-s − 5.76·39-s + 1.56·41-s + 0.304·43-s + 0.447·45-s − 1.16·47-s − 6/7·49-s − 0.840·51-s − 0.549·53-s + 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.377·63-s + ⋯

Functional equation

Λ(s)=(19600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(19600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1960019600    =    24⋅52⋅722^{4} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 1.249711.24971
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 19600, ( :1/2,1/2), 1)(4,\ 19600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9279640351.927964035
L(12)L(\frac12) ≈\approx 1.9279640351.927964035
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−T+T2 1 - T + T^{2}
7C2C_2 1−T+pT2 1 - T + p T^{2}
good3C2C_2 (1−pT+pT2)(1+pT2) ( 1 - p T + p T^{2} )( 1 + p T^{2} )
11C22C_2^2 1−2T−7T2−2pT3+p2T4 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
17C22C_2^2 1+2T−13T2+2pT3+p2T4 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
23C22C_2^2 1−9T+58T2−9pT3+p2T4 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C22C_2^2 1+2T−27T2+2pT3+p2T4 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+8T+27T2+8pT3+p2T4 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
43C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+4T−37T2+4pT3+p2T4 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−8T+5T2−8pT3+p2T4 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+7T−12T2+7pT3+p2T4 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−3T−58T2−3pT3+p2T4 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73C22C_2^2 1+14T+123T2+14pT3+p2T4 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4}
79C2C_2 (1−13T+pT2)(1+17T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
89C22C_2^2 1+13T+80T2+13pT3+p2T4 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−13.60122216544441798159624896179, −12.94793081897094035412996021100, −12.34319715475179651851535376233, −12.26705420246019802564831307460, −11.32006546482095040395029376052, −10.89001663872321024420385947457, −10.02227595596695412511737405192, −9.567992273346978297347743560120, −9.406135851580823636168006377522, −8.696057970109697825872006184581, −8.299185265549282275363671131429, −7.61817640236334635123069039669, −7.04975275008661300422987347886, −6.76744508605909680024719185851, −5.49297177427976641054165719345, −4.86887653197319161164792302111, −4.38522254492250913918981177806, −3.06969582227842589892302745920, −2.73897840237183253770951264479, −1.95938077675247174214824466243, 1.95938077675247174214824466243, 2.73897840237183253770951264479, 3.06969582227842589892302745920, 4.38522254492250913918981177806, 4.86887653197319161164792302111, 5.49297177427976641054165719345, 6.76744508605909680024719185851, 7.04975275008661300422987347886, 7.61817640236334635123069039669, 8.299185265549282275363671131429, 8.696057970109697825872006184581, 9.406135851580823636168006377522, 9.567992273346978297347743560120, 10.02227595596695412511737405192, 10.89001663872321024420385947457, 11.32006546482095040395029376052, 12.26705420246019802564831307460, 12.34319715475179651851535376233, 12.94793081897094035412996021100, 13.60122216544441798159624896179

Graph of the ZZ-function along the critical line