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Properties

Label 4-4050e2-1.1-c1e2-0-16
Degree 44
Conductor 1640250016402500
Sign 11
Analytic cond. 1045.831045.83
Root an. cond. 5.686775.68677
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-s + 16-s + 8·19-s + 18·29-s − 8·31-s − 12·41-s − 2·49-s − 2·61-s − 64-s + 24·71-s − 8·76-s + 32·79-s − 6·89-s + 12·101-s − 22·109-s − 18·116-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 12·164-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s + 1.83·19-s + 3.34·29-s − 1.43·31-s − 1.87·41-s − 2/7·49-s − 0.256·61-s − 1/8·64-s + 2.84·71-s − 0.917·76-s + 3.60·79-s − 0.635·89-s + 1.19·101-s − 2.10·109-s − 1.67·116-s − 2·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.937·164-s + ⋯

Functional equation

Λ(s)=(16402500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16402500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1640250016402500    =    22⋅38⋅542^{2} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1045.831045.83
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16402500, ( :1/2,1/2), 1)(4,\ 16402500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5759889442.575988944
L(12)L(\frac12) ≈\approx 2.5759889442.575988944
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)
bad2C2C_2 1+T2 1 + T^{2}
3 1 1
5 1 1
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C22C_2^2 1−73T2+p2T4 1 - 73 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 T^{2} + 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

−8.568898047053907878572376957310, −8.116066877666922280138122953717, −8.052227604401445531077485653725, −7.66960146563822627235602019293, −7.01268029498040075469767635507, −6.85895891960306442098594212485, −6.44285066467025587181054171199, −6.15985282688146409377175111943, −5.39834659508785558255424498877, −5.25821767358497785555253714993, −4.94375701648568252727273119844, −4.63673060107450965376658497772, −3.91698048710023497064247946627, −3.66355961316262205148517182955, −3.18236732006126239065697112126, −2.85068910397812692199959396366, −2.24789322602034699438186367725, −1.63088216983547463506067505073, −1.01922391665599618006517759318, −0.55821438895858008299415781681, 0.55821438895858008299415781681, 1.01922391665599618006517759318, 1.63088216983547463506067505073, 2.24789322602034699438186367725, 2.85068910397812692199959396366, 3.18236732006126239065697112126, 3.66355961316262205148517182955, 3.91698048710023497064247946627, 4.63673060107450965376658497772, 4.94375701648568252727273119844, 5.25821767358497785555253714993, 5.39834659508785558255424498877, 6.15985282688146409377175111943, 6.44285066467025587181054171199, 6.85895891960306442098594212485, 7.01268029498040075469767635507, 7.66960146563822627235602019293, 8.052227604401445531077485653725, 8.116066877666922280138122953717, 8.568898047053907878572376957310

Graph of the ZZ-function along the critical line