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Properties

Label 4-40e4-1.1-c1e2-0-27
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
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  + 6·9-s + 8·11-s − 8·19-s − 4·29-s + 16·31-s − 12·41-s − 2·49-s + 8·59-s + 4·61-s + 27·81-s + 12·89-s + 48·99-s − 12·101-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 + 22·169-s − 48·171-s + 173-s + ⋯
L(s)  = 1  + 2·9-s + 2.41·11-s − 1.83·19-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 2/7·49-s + 1.04·59-s + 0.512·61-s + 3·81-s + 1.27·89-s + 4.82·99-s − 1.19·101-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 + 1.69·169-s − 3.67·171-s + 0.0760·173-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5255686263.525568626
L(12)L(\frac12) ≈\approx 3.5255686263.525568626
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
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
37C22C_2^2 1−38T2+p2T4 1 - 38 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 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C22C_2^2 1+2T2+p2T4 1 + 2 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

−9.524562819225867659361458405960, −9.363805917093392064592456493400, −8.754636530346371975029320183850, −8.474820321447292991901261159846, −8.140500694834928447032980225688, −7.53915116080628948119860829468, −7.04271225572366521702658589920, −6.69296861528944708464781370553, −6.50688668432075950827878856500, −6.26692918769151761124674564637, −5.57325222919881844515671315446, −4.75868723536041397725577008107, −4.51510164471184856655057398001, −4.27042601855627920301222850342, −3.63474014915657021108151792694, −3.50313101214119214995756591303, −2.41853587344961721591346306465, −1.88216750789001048552953637940, −1.40211422804760138264466826050, −0.814050284895622139745909516269, 0.814050284895622139745909516269, 1.40211422804760138264466826050, 1.88216750789001048552953637940, 2.41853587344961721591346306465, 3.50313101214119214995756591303, 3.63474014915657021108151792694, 4.27042601855627920301222850342, 4.51510164471184856655057398001, 4.75868723536041397725577008107, 5.57325222919881844515671315446, 6.26692918769151761124674564637, 6.50688668432075950827878856500, 6.69296861528944708464781370553, 7.04271225572366521702658589920, 7.53915116080628948119860829468, 8.140500694834928447032980225688, 8.474820321447292991901261159846, 8.754636530346371975029320183850, 9.363805917093392064592456493400, 9.524562819225867659361458405960

Graph of the ZZ-function along the critical line