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Properties

Label 8-570e4-1.1-c3e4-0-1
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.27927\times 10^{6}$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·3-s + 4·4-s + 10·5-s + 24·6-s + 6·7-s − 16·8-s + 9·9-s + 40·10-s + 6·11-s + 24·12-s + 109·13-s + 24·14-s + 60·15-s − 64·16-s − 6·17-s + 36·18-s − 19-s + 40·20-s + 36·21-s + 24·22-s + 58·23-s − 96·24-s + 25·25-s + 436·26-s − 54·27-s + 24·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s + 0.323·7-s − 0.707·8-s + 1/3·9-s + 1.26·10-s + 0.164·11-s + 0.577·12-s + 2.32·13-s + 0.458·14-s + 1.03·15-s − 16-s − 0.0856·17-s + 0.471·18-s − 0.0120·19-s + 0.447·20-s + 0.374·21-s + 0.232·22-s + 0.525·23-s − 0.816·24-s + 1/5·25-s + 3.28·26-s − 0.384·27-s + 0.161·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.27927\times 10^{6}\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.449425067\)
\(L(\frac12)\) \(\approx\) \(4.449425067\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 + T - 702 p T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
good7$D_{4}$ \( ( 1 - 3 T + 568 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 3 T + 1582 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 109 T + 4637 T^{2} - 310650 T^{3} + 21546170 T^{4} - 310650 p^{3} T^{5} + 4637 p^{6} T^{6} - 109 p^{9} T^{7} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 + 6 T + 2226 T^{2} - 72096 T^{3} - 19518337 T^{4} - 72096 p^{3} T^{5} + 2226 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 58 T - 21330 T^{2} - 20880 T^{3} + 420827959 T^{4} - 20880 p^{3} T^{5} - 21330 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 307 T + 22991 T^{2} + 6901360 T^{3} + 2184240382 T^{4} + 6901360 p^{3} T^{5} + 22991 p^{6} T^{6} + 307 p^{9} T^{7} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 - 376 T + 94445 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 263 T + 117516 T^{2} + 263 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 422 T + 50 T^{2} + 16961024 T^{3} + 14672273551 T^{4} + 16961024 p^{3} T^{5} + 50 p^{6} T^{6} + 422 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 - 79 T - 151327 T^{2} + 114234 T^{3} + 18010108388 T^{4} + 114234 p^{3} T^{5} - 151327 p^{6} T^{6} - 79 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 390 T + 14654 T^{2} + 27378000 T^{3} - 7679290713 T^{4} + 27378000 p^{3} T^{5} + 14654 p^{6} T^{6} - 390 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 60 T - 225790 T^{2} - 4101840 T^{3} + 29919453771 T^{4} - 4101840 p^{3} T^{5} - 225790 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 209 T - 368257 T^{2} + 4180 p T^{3} + 34053784 p^{2} T^{4} + 4180 p^{4} T^{5} - 368257 p^{6} T^{6} + 209 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 944 T + 272591 T^{2} + 155366352 T^{3} + 128806602248 T^{4} + 155366352 p^{3} T^{5} + 272591 p^{6} T^{6} + 944 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 89 T - 595465 T^{2} - 165540 T^{3} + 271233939104 T^{4} - 165540 p^{3} T^{5} - 595465 p^{6} T^{6} - 89 p^{9} T^{7} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 - 537 T - 256039 T^{2} + 92049318 T^{3} + 65069349384 T^{4} + 92049318 p^{3} T^{5} - 256039 p^{6} T^{6} - 537 p^{9} T^{7} + p^{12} T^{8} \)
73$D_4\times C_2$ \( 1 - 233 T - 514975 T^{2} + 48643410 T^{3} + 151607931494 T^{4} + 48643410 p^{3} T^{5} - 514975 p^{6} T^{6} - 233 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 1880 T + 1838363 T^{2} + 1334722920 T^{3} + 890857008248 T^{4} + 1334722920 p^{3} T^{5} + 1838363 p^{6} T^{6} + 1880 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 274 T + 757822 T^{2} + 274 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 699 T - 624897 T^{2} - 207211560 T^{3} + 398447946674 T^{4} - 207211560 p^{3} T^{5} - 624897 p^{6} T^{6} + 699 p^{9} T^{7} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 + 422 T - 1668214 T^{2} + 91152 p T^{3} + 256489727 p^{2} T^{4} + 91152 p^{4} T^{5} - 1668214 p^{6} T^{6} + 422 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.38518086213171622048073725682, −7.00168075723711761472781543734, −6.59066030904246029191251555719, −6.49100134590152023275551099610, −6.32663326010342467896182249638, −6.07168185043306979992381906608, −5.92719319620837510607130352787, −5.52850133254690730597333454133, −5.26442707020897910192610934650, −5.15941912203602927681446264951, −4.67760183437504803775880343977, −4.63516778648210098947533727171, −4.28279320095921610477079682228, −3.93994960305124778699663623759, −3.75265531057865727235587906633, −3.35607680575835480389713233441, −3.15460403228478280247723626745, −3.12459565395511515748330159295, −2.57068449770902692292624264936, −2.43375007685361610707691196924, −1.78563813383805242847849408729, −1.50083571953796585349838885039, −1.43294179842841494333920820810, −0.941221978153201612701785401446, −0.15727179410324322326804221295, 0.15727179410324322326804221295, 0.941221978153201612701785401446, 1.43294179842841494333920820810, 1.50083571953796585349838885039, 1.78563813383805242847849408729, 2.43375007685361610707691196924, 2.57068449770902692292624264936, 3.12459565395511515748330159295, 3.15460403228478280247723626745, 3.35607680575835480389713233441, 3.75265531057865727235587906633, 3.93994960305124778699663623759, 4.28279320095921610477079682228, 4.63516778648210098947533727171, 4.67760183437504803775880343977, 5.15941912203602927681446264951, 5.26442707020897910192610934650, 5.52850133254690730597333454133, 5.92719319620837510607130352787, 6.07168185043306979992381906608, 6.32663326010342467896182249638, 6.49100134590152023275551099610, 6.59066030904246029191251555719, 7.00168075723711761472781543734, 7.38518086213171622048073725682

Graph of the $Z$-function along the critical line