↑

Properties

Label 4-1920e2-1.1-c1e2-0-10
Degree 44
Conductor 36864003686400
Sign 11
Analytic cond. 235.048235.048
Root an. cond. 3.915513.91551
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 9-s − 12·19-s + 11·25-s + 16·29-s − 16·31-s − 12·41-s − 4·45-s − 2·49-s + 24·59-s + 28·61-s − 16·71-s + 24·79-s + 81-s − 12·89-s − 48·95-s + 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s − 64·155-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 2.75·19-s + 11/5·25-s + 2.97·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s − 2/7·49-s + 3.12·59-s + 3.58·61-s − 1.89·71-s + 2.70·79-s + 1/9·81-s − 1.27·89-s − 4.92·95-s + 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯

Functional equation

Λ(s)=(3686400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3686400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 36864003686400    =    214⋅32⋅522^{14} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 235.048235.048
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3686400, ( :1/2,1/2), 1)(4,\ 3686400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7417405582.741740558
L(12)L(\frac12) ≈\approx 2.7417405582.741740558
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+T2 1 + T^{2}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + 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+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{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

−9.406171627091146240017670032624, −8.852788276641653995601695500925, −8.677837639968309451837005472450, −8.392865466590526959443821282965, −8.121732002480809353675107529992, −7.21516605585194194959190067794, −6.79989865345280484695001224396, −6.62294726370191494405706318762, −6.37911386646073402901520764011, −5.63860816953378232698023246273, −5.57182273236609865220494082500, −4.95065225135551634082598349079, −4.73602281260995820595505859434, −3.83291737925194375357772197923, −3.76119844539937844423297141054, −2.79139402660676206332875153889, −2.39645388668815366288858194558, −2.05373433774555767093572023145, −1.52897513463252493945054421534, −0.58184425971407767233654672679, 0.58184425971407767233654672679, 1.52897513463252493945054421534, 2.05373433774555767093572023145, 2.39645388668815366288858194558, 2.79139402660676206332875153889, 3.76119844539937844423297141054, 3.83291737925194375357772197923, 4.73602281260995820595505859434, 4.95065225135551634082598349079, 5.57182273236609865220494082500, 5.63860816953378232698023246273, 6.37911386646073402901520764011, 6.62294726370191494405706318762, 6.79989865345280484695001224396, 7.21516605585194194959190067794, 8.121732002480809353675107529992, 8.392865466590526959443821282965, 8.677837639968309451837005472450, 8.852788276641653995601695500925, 9.406171627091146240017670032624

Graph of the ZZ-function along the critical line