↑

Properties

Label 4-9200e2-1.1-c1e2-0-13
Degree 44
Conductor 8464000084640000
Sign 11
Analytic cond. 5396.715396.71
Root an. cond. 8.571018.57101
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·7-s + 2·9-s − 2·11-s + 4·17-s − 2·19-s − 8·21-s + 2·23-s − 6·27-s − 8·29-s + 8·31-s + 4·33-s + 18·37-s + 6·41-s − 14·43-s − 10·47-s + 3·49-s − 8·51-s − 2·53-s + 4·57-s − 2·59-s − 10·61-s + 8·63-s − 8·67-s − 4·69-s − 20·71-s − 10·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.970·17-s − 0.458·19-s − 1.74·21-s + 0.417·23-s − 1.15·27-s − 1.48·29-s + 1.43·31-s + 0.696·33-s + 2.95·37-s + 0.937·41-s − 2.13·43-s − 1.45·47-s + 3/7·49-s − 1.12·51-s − 0.274·53-s + 0.529·57-s − 0.260·59-s − 1.28·61-s + 1.00·63-s − 0.977·67-s − 0.481·69-s − 2.37·71-s − 1.17·73-s + ⋯

Functional equation

Λ(s)=(84640000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(84640000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8464000084640000    =    28⋅54⋅2322^{8} \cdot 5^{4} \cdot 23^{2}
Sign: 11
Analytic conductor: 5396.715396.71
Root analytic conductor: 8.571018.57101
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 84640000, ( :1/2,1/2), 1)(4,\ 84640000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
23C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
7D4D_{4} 1−4T+13T2−4pT3+p2T4 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4}
11C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
13C22C_2^2 1+21T2+p2T4 1 + 21 T^{2} + p^{2} T^{4}
17D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
29D4D_{4} 1+8T+69T2+8pT3+p2T4 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−8T+58T2−8pT3+p2T4 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−18T+150T2−18pT3+p2T4 1 - 18 T + 150 T^{2} - 18 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+11T2−6pT3+p2T4 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+14T+115T2+14pT3+p2T4 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+10T+74T2+10pT3+p2T4 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+2T+102T2+2pT3+p2T4 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+2T+114T2+2pT3+p2T4 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+10T+142T2+10pT3+p2T4 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+8T+70T2+8pT3+p2T4 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}
71C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
73D4D_{4} 1+10T+151T2+10pT3+p2T4 1 + 10 T + 151 T^{2} + 10 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+49T2+8pT3+p2T4 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
89C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97D4D_{4} 1+10T+94T2+10pT3+p2T4 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4}
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

−7.59465090594354315121849235360, −7.54318682309951755788940437998, −6.62487848911350690130209658946, −6.58199201198977058359789586536, −6.10059843722885251835990738507, −5.72959658414275361316010104606, −5.45131220258259683041189229071, −5.34584069501345950075555752183, −4.61512916477133518541953882121, −4.46185698311575780337723744245, −4.42578066512789798888497954002, −3.78016847130066560148602255163, −3.07031978392638603524386619879, −2.92638183571929890514770699797, −2.39976234700353712294350703988, −1.71246928549210154885746249595, −1.33977497191023249185908803634, −1.20294161615390550317929615551, 0, 0, 1.20294161615390550317929615551, 1.33977497191023249185908803634, 1.71246928549210154885746249595, 2.39976234700353712294350703988, 2.92638183571929890514770699797, 3.07031978392638603524386619879, 3.78016847130066560148602255163, 4.42578066512789798888497954002, 4.46185698311575780337723744245, 4.61512916477133518541953882121, 5.34584069501345950075555752183, 5.45131220258259683041189229071, 5.72959658414275361316010104606, 6.10059843722885251835990738507, 6.58199201198977058359789586536, 6.62487848911350690130209658946, 7.54318682309951755788940437998, 7.59465090594354315121849235360

Graph of the ZZ-function along the critical line