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Properties

Label 4-1440e2-1.1-c1e2-0-38
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 132.214132.214
Root an. cond. 3.390933.39093
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 − 3·7-s + 6·9-s + 4·11-s + 4·13-s − 3·15-s + 4·17-s + 4·19-s − 9·21-s + 7·23-s + 9·27-s + 9·29-s − 6·31-s + 12·33-s + 3·35-s + 4·37-s + 12·39-s − 9·41-s − 4·43-s − 6·45-s + 3·47-s + 7·49-s + 12·51-s + 12·53-s − 4·55-s + 12·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s + 1.20·11-s + 1.10·13-s − 0.774·15-s + 0.970·17-s + 0.917·19-s − 1.96·21-s + 1.45·23-s + 1.73·27-s + 1.67·29-s − 1.07·31-s + 2.08·33-s + 0.507·35-s + 0.657·37-s + 1.92·39-s − 1.40·41-s − 0.609·43-s − 0.894·45-s + 0.437·47-s + 49-s + 1.68·51-s + 1.64·53-s − 0.539·55-s + 1.58·57-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 132.214132.214
Root analytic conductor: 3.390933.39093
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :1/2,1/2), 1)(4,\ 2073600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.1887358395.188735839
L(12)L(\frac12) ≈\approx 5.1887358395.188735839
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
3C2C_2 1−pT+pT2 1 - p T + p T^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
good7C22C_2^2 1+3T+2T2+3pT3+p2T4 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−4T+5T2−4pT3+p2T4 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−4T+3T2−4pT3+p2T4 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
23C22C_2^2 1−7T+26T2−7pT3+p2T4 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−9T+52T2−9pT3+p2T4 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+6T+5T2+6pT3+p2T4 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C22C_2^2 1+9T+40T2+9pT3+p2T4 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+4T−27T2+4pT3+p2T4 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−3T−38T2−3pT3+p2T4 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1−6T−23T2−6pT3+p2T4 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−14T+pT2)(1+T+pT2) ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} )
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+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
79C22C_2^2 1+10T+21T2+10pT3+p2T4 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+7T−34T2+7pT3+p2T4 1 + 7 T - 34 T^{2} + 7 p T^{3} + p^{2} T^{4}
89C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
97C2C_2 (1−19T+pT2)(1+5T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{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

−9.711676928177112821308923450659, −9.286516392215275375626881020447, −8.820509884844381138709402753007, −8.684205340604873195326364701413, −8.077001323888815233106077750829, −8.034347358198727872612474359148, −7.09647897740824745320782206073, −7.03247170236136401176869984337, −6.77301527900561703522936967243, −6.20910172218835029345130070436, −5.48529482331468642891240993658, −5.25545604249761292761246106319, −4.34169950456451886517474604303, −3.99332336757437231258778983069, −3.49435726826145050355321227370, −3.36095979141425679052665092176, −2.84595659644318422784676185891, −2.27032257874278009754569112778, −1.24681582128021905837638373273, −1.02074679397857580838969339398, 1.02074679397857580838969339398, 1.24681582128021905837638373273, 2.27032257874278009754569112778, 2.84595659644318422784676185891, 3.36095979141425679052665092176, 3.49435726826145050355321227370, 3.99332336757437231258778983069, 4.34169950456451886517474604303, 5.25545604249761292761246106319, 5.48529482331468642891240993658, 6.20910172218835029345130070436, 6.77301527900561703522936967243, 7.03247170236136401176869984337, 7.09647897740824745320782206073, 8.034347358198727872612474359148, 8.077001323888815233106077750829, 8.684205340604873195326364701413, 8.820509884844381138709402753007, 9.286516392215275375626881020447, 9.711676928177112821308923450659

Graph of the ZZ-function along the critical line