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Properties

Label 8-20e8-1.1-c5e4-0-5
Degree 88
Conductor 2560000000025600000000
Sign 11
Analytic cond. 1.69387×1071.69387\times 10^{7}
Root an. cond. 8.009588.00958
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 660·13-s − 5.06e3·17-s − 5.54e3·37-s + 8.71e3·41-s + 9.56e4·53-s − 1.43e5·61-s + 8.64e4·73-s + 8.71e4·81-s + 2.46e3·97-s + 5.87e5·101-s − 4.68e4·113-s + 6.13e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.17e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 1.08·13-s − 4.24·17-s − 0.665·37-s + 0.809·41-s + 4.67·53-s − 4.93·61-s + 1.89·73-s + 1.47·81-s + 0.0265·97-s + 5.73·101-s − 0.344·113-s + 3.80·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.586·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯

Functional equation

Λ(s)=((216⋅58)s/2 ΓC(s)4 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((216⋅58)s/2 ΓC(s+5/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅582^{16} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.69387×1071.69387\times 10^{7}
Root analytic conductor: 8.009588.00958
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅58, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{16} \cdot 5^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 3.4796023743.479602374
L(12)L(\frac12) ≈\approx 3.4796023743.479602374
L(72)L(\frac{7}{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
good3C23C_2^3 1−87122T4+p20T8 1 - 87122 T^{4} + p^{20} T^{8}
7C23C_2^3 1−549771682T4+p20T8 1 - 549771682 T^{4} + p^{20} T^{8}
11C22C_2^2 (1−306602T2+p10T4)2 ( 1 - 306602 T^{2} + p^{10} T^{4} )^{2}
13C22C_2^2 (1+330T+54450T2+330p5T3+p10T4)2 ( 1 + 330 T + 54450 T^{2} + 330 p^{5} T^{3} + p^{10} T^{4} )^{2}
17C22C_2^2 (1+2530T+3200450T2+2530p5T3+p10T4)2 ( 1 + 2530 T + 3200450 T^{2} + 2530 p^{5} T^{3} + p^{10} T^{4} )^{2}
19C22C_2^2 (1−2549802T2+p10T4)2 ( 1 - 2549802 T^{2} + p^{10} T^{4} )^{2}
23C23C_2^3 1+68424579426718T4+p20T8 1 + 68424579426718 T^{4} + p^{20} T^{8}
29C22C_2^2 (1−34283082T2+p10T4)2 ( 1 - 34283082 T^{2} + p^{10} T^{4} )^{2}
31C22C_2^2 (1−222542pT2+p10T4)2 ( 1 - 222542 p T^{2} + p^{10} T^{4} )^{2}
37C22C_2^2 (1+2770T+3836450T2+2770p5T3+p10T4)2 ( 1 + 2770 T + 3836450 T^{2} + 2770 p^{5} T^{3} + p^{10} T^{4} )^{2}
41C2C_2 (1−2178T+p5T2)4 ( 1 - 2178 T + p^{5} T^{2} )^{4}
43C23C_2^3 1−35368995141923122T4+p20T8 1 - 35368995141923122 T^{4} + p^{20} T^{8}
47C23C_2^3 1+62763367693678078T4+p20T8 1 + 62763367693678078 T^{4} + p^{20} T^{8}
53C22C_2^2 (1−47830T+1143854450T2−47830p5T3+p10T4)2 ( 1 - 47830 T + 1143854450 T^{2} - 47830 p^{5} T^{3} + p^{10} T^{4} )^{2}
59C22C_2^2 (1+693226598T2+p10T4)2 ( 1 + 693226598 T^{2} + p^{10} T^{4} )^{2}
61C2C_2 (1+35882T+p5T2)4 ( 1 + 35882 T + p^{5} T^{2} )^{4}
67C23C_2^3 1−2056557036860651602T4+p20T8 1 - 2056557036860651602 T^{4} + p^{20} T^{8}
71C22C_2^2 (1+894616798T2+p10T4)2 ( 1 + 894616798 T^{2} + p^{10} T^{4} )^{2}
73C22C_2^2 (1−43230T+934416450T2−43230p5T3+p10T4)2 ( 1 - 43230 T + 934416450 T^{2} - 43230 p^{5} T^{3} + p^{10} T^{4} )^{2}
79C22C_2^2 (1+5673984798T2+p10T4)2 ( 1 + 5673984798 T^{2} + p^{10} T^{4} )^{2}
83C23C_2^3 1−7276763020942840082T4+p20T8 1 - 7276763020942840082 T^{4} + p^{20} T^{8}
89C22C_2^2 (1+1924732878T2+p10T4)2 ( 1 + 1924732878 T^{2} + p^{10} T^{4} )^{2}
97C22C_2^2 (1−1230T+756450T2−1230p5T3+p10T4)2 ( 1 - 1230 T + 756450 T^{2} - 1230 p^{5} T^{3} + p^{10} T^{4} )^{2}
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   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

−7.23316512281745283489232443903, −6.98405665870692257804712361159, −6.91361833754589925361790239110, −6.54883129014935147662487027109, −6.49204734186441541860349634935, −5.89825983575944649431321144987, −5.86277668950416943455681574677, −5.73481602893949889458225910964, −5.15064703135633514759530460085, −4.80287331770110658221778125052, −4.70309250929625057908708920339, −4.41015798840778967627552723889, −4.35340241105729351608377147347, −4.04465533982118009816183851745, −3.48510024829449422319992825095, −3.19149509949423687517392965240, −3.10075915194474378097827084069, −2.32527892251211426172158101898, −2.17291549402563526941784270285, −2.17287152481114052290610149932, −1.98989794737957290897775853581, −1.31295783682318706234672943366, −0.70130318991100519062555050103, −0.41728810129350140301967538304, −0.39658871895481923981925215144, 0.39658871895481923981925215144, 0.41728810129350140301967538304, 0.70130318991100519062555050103, 1.31295783682318706234672943366, 1.98989794737957290897775853581, 2.17287152481114052290610149932, 2.17291549402563526941784270285, 2.32527892251211426172158101898, 3.10075915194474378097827084069, 3.19149509949423687517392965240, 3.48510024829449422319992825095, 4.04465533982118009816183851745, 4.35340241105729351608377147347, 4.41015798840778967627552723889, 4.70309250929625057908708920339, 4.80287331770110658221778125052, 5.15064703135633514759530460085, 5.73481602893949889458225910964, 5.86277668950416943455681574677, 5.89825983575944649431321144987, 6.49204734186441541860349634935, 6.54883129014935147662487027109, 6.91361833754589925361790239110, 6.98405665870692257804712361159, 7.23316512281745283489232443903

Graph of the ZZ-function along the critical line