↑

Properties

Label 8-7200e4-1.1-c1e4-0-3
Degree 88
Conductor 2.687×10152.687\times 10^{15}
Sign 11
Analytic cond. 1.09254×1071.09254\times 10^{7}
Root an. cond. 7.582367.58236
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  − 20·41-s − 12·49-s + 40·61-s − 20·89-s + 8·101-s − 24·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 3.12·41-s − 1.71·49-s + 5.12·61-s − 2.11·89-s + 0.796·101-s − 2.29·109-s − 3.09·121-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 + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

Λ(s)=((220⋅38⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅38⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅38⋅582^{20} \cdot 3^{8} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.09254×1071.09254\times 10^{7}
Root analytic conductor: 7.582367.58236
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅38⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.33058238250.3305823825
L(12)L(\frac12) ≈\approx 0.33058238250.3305823825
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
3 1 1
5 1 1
good7C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1+17T2+p2T4)2 ( 1 + 17 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−6T+pT2)2(1+6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}
17C22C_2^2 (1+15T2+p2T4)2 ( 1 + 15 T^{2} + p^{2} T^{4} )^{2}
19C22C_2^2 (1−7T2+p2T4)2 ( 1 - 7 T^{2} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−26T2+p2T4)2 ( 1 - 26 T^{2} + p^{2} T^{4} )^{2}
29C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
31C22C_2^2 (1+42T2+p2T4)2 ( 1 + 42 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41C2C_2 (1+5T+pT2)4 ( 1 + 5 T + p T^{2} )^{4}
43C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
47C22C_2^2 (1−14T2+p2T4)2 ( 1 - 14 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+38T2+p2T4)2 ( 1 + 38 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
67C22C_2^2 (1−129T2+p2T4)2 ( 1 - 129 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+62T2+p2T4)2 ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−65T2+p2T4)2 ( 1 - 65 T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1+138T2+p2T4)2 ( 1 + 138 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−41T2+p2T4)2 ( 1 - 41 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1+5T+pT2)4 ( 1 + 5 T + p T^{2} )^{4}
97C22C_2^2 (1−190T2+p2T4)2 ( 1 - 190 T^{2} + p^{2} T^{4} )^{2}
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

−5.42203298465228833083737670731, −5.39784149886645335350533202719, −5.26463914749291873568210757351, −5.01829961303281057425261312630, −4.83719862463972772640668932897, −4.82164592941250853059079769600, −4.29865035123383011571593854429, −4.26128975382218267191807533246, −3.99259519659740165898230981111, −3.91440677537429529658819380054, −3.65561792616361294338861589601, −3.44713520898222059297840299683, −3.26124728183219900421952942671, −3.12631947315584158575051198782, −2.85720539381368682477026792647, −2.46638329762792549391810432918, −2.35704000147278000255294995888, −2.30861678740120204224271111606, −1.88520164164897844474793435139, −1.73422867820027683031652218276, −1.29348053404205404734656024124, −1.12233395391731144027394747420, −1.12130258632548584779506121241, −0.43061105971833427041287858923, −0.087654335613115036668467431522, 0.087654335613115036668467431522, 0.43061105971833427041287858923, 1.12130258632548584779506121241, 1.12233395391731144027394747420, 1.29348053404205404734656024124, 1.73422867820027683031652218276, 1.88520164164897844474793435139, 2.30861678740120204224271111606, 2.35704000147278000255294995888, 2.46638329762792549391810432918, 2.85720539381368682477026792647, 3.12631947315584158575051198782, 3.26124728183219900421952942671, 3.44713520898222059297840299683, 3.65561792616361294338861589601, 3.91440677537429529658819380054, 3.99259519659740165898230981111, 4.26128975382218267191807533246, 4.29865035123383011571593854429, 4.82164592941250853059079769600, 4.83719862463972772640668932897, 5.01829961303281057425261312630, 5.26463914749291873568210757351, 5.39784149886645335350533202719, 5.42203298465228833083737670731

Graph of the ZZ-function along the critical line