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Properties

Label 4-800e2-1.1-c1e2-0-33
Degree 44
Conductor 640000640000
Sign 11
Analytic cond. 40.806940.8069
Root an. cond. 2.527452.52745
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  + 2·9-s + 8·11-s + 16·19-s + 4·29-s − 8·31-s − 20·41-s + 10·49-s + 4·61-s + 24·71-s − 16·79-s − 5·81-s + 12·89-s + 16·99-s + 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 32·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.41·11-s + 3.67·19-s + 0.742·29-s − 1.43·31-s − 3.12·41-s + 10/7·49-s + 0.512·61-s + 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.60·99-s + 2.78·101-s + 2.68·109-s + 2.36·121-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 + 0.0773·167-s − 0.769·169-s + 2.44·171-s + ⋯

Functional equation

Λ(s)=(640000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(640000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 640000640000    =    210⋅542^{10} \cdot 5^{4}
Sign: 11
Analytic conductor: 40.806940.8069
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 640000, ( :1/2,1/2), 1)(4,\ 640000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9721095432.972109543
L(12)L(\frac12) ≈\approx 2.9721095432.972109543
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
5 1 1
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 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

−10.07504478878799993589527893617, −10.07494148824040044632683123428, −9.754399181016294135641612742267, −9.147593313744008898440551813804, −8.879208929021368555566973043790, −8.589731752532433791165519710500, −7.69092572792073251787856259954, −7.44420438790112111466250655457, −7.02180447652963304443323578739, −6.71305447448958984361620162138, −6.15556154248036396710764626554, −5.61394579599834384622783529656, −4.96530917422889623830417257180, −4.86898791580292905415471260025, −3.74305977784943563899566738819, −3.67468791370936355590713059598, −3.28131717218369490551418823091, −2.21195462598258886970323004006, −1.30114715274294811059575399550, −1.11332336049578327972894929450, 1.11332336049578327972894929450, 1.30114715274294811059575399550, 2.21195462598258886970323004006, 3.28131717218369490551418823091, 3.67468791370936355590713059598, 3.74305977784943563899566738819, 4.86898791580292905415471260025, 4.96530917422889623830417257180, 5.61394579599834384622783529656, 6.15556154248036396710764626554, 6.71305447448958984361620162138, 7.02180447652963304443323578739, 7.44420438790112111466250655457, 7.69092572792073251787856259954, 8.589731752532433791165519710500, 8.879208929021368555566973043790, 9.147593313744008898440551813804, 9.754399181016294135641612742267, 10.07494148824040044632683123428, 10.07504478878799993589527893617

Graph of the ZZ-function along the critical line