↑

Properties

Label 4-32000-1.1-c1e2-0-3
Degree 44
Conductor 3200032000
Sign 11
Analytic cond. 2.040342.04034
Root an. cond. 1.195151.19515
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·9-s + 25-s + 12·29-s − 12·41-s + 2·45-s + 10·49-s − 4·61-s − 5·81-s − 12·89-s + 12·101-s − 4·109-s − 22·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.447·5-s + 2/3·9-s + 1/5·25-s + 2.22·29-s − 1.87·41-s + 0.298·45-s + 10/7·49-s − 0.512·61-s − 5/9·81-s − 1.27·89-s + 1.19·101-s − 0.383·109-s − 2·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-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 + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=(32000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(32000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3200032000    =    28⋅532^{8} \cdot 5^{3}
Sign: 11
Analytic conductor: 2.040342.04034
Root analytic conductor: 1.195151.19515
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 32000, ( :1/2,1/2), 1)(4,\ 32000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4362478511.436247851
L(12)L(\frac12) ≈\approx 1.4362478511.436247851
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
5C1C_1 1−T 1 - T
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+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
47C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
show more
show less
   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

−10.33788580631742990536902027000, −10.05879041470947803793456960886, −9.522648273985201214197269118626, −8.796884726322918039678921570591, −8.479565110616274416699778327122, −7.81615254862260954513085168025, −7.10947840203714705149389544521, −6.69688321114788058924654444075, −6.13072415585895077395132843521, −5.37979552590376931141997700211, −4.77253387895635684692535129760, −4.16626097376329934356101731495, −3.25546128645228249942275343375, −2.42787059732878662273347800550, −1.31802743791612034341244693514, 1.31802743791612034341244693514, 2.42787059732878662273347800550, 3.25546128645228249942275343375, 4.16626097376329934356101731495, 4.77253387895635684692535129760, 5.37979552590376931141997700211, 6.13072415585895077395132843521, 6.69688321114788058924654444075, 7.10947840203714705149389544521, 7.81615254862260954513085168025, 8.479565110616274416699778327122, 8.796884726322918039678921570591, 9.522648273985201214197269118626, 10.05879041470947803793456960886, 10.33788580631742990536902027000

Graph of the ZZ-function along the critical line