↑

Properties

Label 8-2400e4-1.1-c0e4-0-1
Degree 88
Conductor 3.318×10133.318\times 10^{13}
Sign 11
Analytic cond. 2.058132.05813
Root an. cond. 1.094421.09442
Motivic weight 00
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  − 81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅34⋅582^{20} \cdot 3^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 2.058132.05813
Root analytic conductor: 1.094421.09442
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅34⋅58, ( :0,0,0,0), 1)(8,\ 2^{20} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1014061861.101406186
L(12)L(\frac12) ≈\approx 1.1014061861.101406186
L(1)L(1) 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
3C22C_2^2 1+T4 1 + T^{4}
5 1 1
good7C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
11C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
13C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
17C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
19C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
23C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
29C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
31C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
37C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
41C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
43C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
47C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
53C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
59C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
61C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
67C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
71C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
73C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
79C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
83C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
show more
show less
   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.60154330534272279093343668521, −6.15258798723087000736068013009, −6.05318746570498503114681116315, −5.96399062176313524658500968717, −5.94233026105446575305931767642, −5.61195730519488714905864088034, −5.18001707253011900970451449656, −5.13959122435953503910722904017, −4.85542105385135321674884481014, −4.68960534326432404976060852232, −4.54893905190806380733199359028, −4.15376682563979416024946002433, −4.15350253278408281764083778630, −3.58237198836779789491750069634, −3.50511097719140906810486465285, −3.43716643650274088073445852885, −3.21863842390714793085334582859, −2.55292450455394578900997986732, −2.52685950090802457999639509183, −2.45977424059046015242562918504, −2.07378983996711400094650400249, −1.60592898770165627623366021782, −1.34262124958448198976749591894, −1.17123296346822384525524528466, −0.49538494907228287729740016498, 0.49538494907228287729740016498, 1.17123296346822384525524528466, 1.34262124958448198976749591894, 1.60592898770165627623366021782, 2.07378983996711400094650400249, 2.45977424059046015242562918504, 2.52685950090802457999639509183, 2.55292450455394578900997986732, 3.21863842390714793085334582859, 3.43716643650274088073445852885, 3.50511097719140906810486465285, 3.58237198836779789491750069634, 4.15350253278408281764083778630, 4.15376682563979416024946002433, 4.54893905190806380733199359028, 4.68960534326432404976060852232, 4.85542105385135321674884481014, 5.13959122435953503910722904017, 5.18001707253011900970451449656, 5.61195730519488714905864088034, 5.94233026105446575305931767642, 5.96399062176313524658500968717, 6.05318746570498503114681116315, 6.15258798723087000736068013009, 6.60154330534272279093343668521

Graph of the ZZ-function along the critical line