Properties

Label 8-320e4-1.1-c8e4-0-0
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $2.88797\times 10^{8}$
Root an. cond. $11.4175$
Motivic weight $8$
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  − 100·5-s + 3.87e3·9-s − 7.73e5·25-s − 4.16e6·29-s − 3.32e6·41-s − 3.87e5·45-s − 2.15e7·49-s − 4.24e7·61-s − 7.48e7·81-s + 3.22e8·89-s − 3.87e8·101-s + 2.20e8·109-s + 3.87e8·121-s + 1.16e8·125-s + 127-s + 131-s + 137-s + 139-s + 4.16e8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.52e9·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 0.159·5-s + 0.590·9-s − 1.98·25-s − 5.89·29-s − 1.17·41-s − 0.0945·45-s − 3.74·49-s − 3.06·61-s − 1.73·81-s + 5.14·89-s − 3.71·101-s + 1.56·109-s + 1.80·121-s + 0.477·125-s + 0.942·145-s + 3.09·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.88797\times 10^{8}\)
Root analytic conductor: \(11.4175\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 4, 4, 4, 4 ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.1478786521\)
\(L(\frac12)\) \(\approx\) \(0.1478786521\)
\(L(5)\) 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 \( 1 \)
5$C_2$ \( ( 1 + 2 p^{2} T + p^{8} T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 646 p T^{2} + p^{16} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 220238 p^{2} T^{2} + p^{16} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 193781762 T^{2} + p^{16} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 97037834 p T^{2} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 6043188482 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 5539870082 T^{2} + p^{16} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 142069658222 T^{2} + p^{16} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 1041922 T + p^{8} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 487873850882 T^{2} + p^{16} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 3082115005442 T^{2} + p^{16} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 831982 T + p^{8} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 16588024818542 T^{2} + p^{16} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 7011460446062 T^{2} + p^{16} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 7109779173122 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 63865039952642 T^{2} + p^{16} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 10617778 T + p^{8} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 583910624435218 T^{2} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 207543434244478 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 1183233014770562 T^{2} + p^{16} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 336865887770878 T^{2} + p^{16} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 3731242433596142 T^{2} + p^{16} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 80736322 T + p^{8} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 3474443214251522 T^{2} + p^{16} T^{4} )^{2} \)
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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

−7.00083443915422023261798862156, −6.96858691668564559204308352713, −6.34172043198974312739762736923, −6.23757582369929073334200856515, −6.07582074538987587300712214949, −5.75868635530745618459268134434, −5.41071178800638831500692382017, −5.32713873338853142455431870390, −4.92676933425425517985232750969, −4.73269767142320120667910626329, −4.26805590734628580262250456779, −4.12217912254924503675000079622, −3.77472306197706415176602451654, −3.54535807012569873530300333170, −3.42082938970506115434798711658, −2.99836881175627482680437891435, −2.82754981782417286067803685224, −1.99300006853509246466736611099, −1.90905473831419569390846587109, −1.85655848119167621908075159112, −1.59476994122708884207648717350, −1.38583430836009864903264407876, −0.58854728317271444063581659187, −0.40291360473737722197930117533, −0.05872165359954467966744972753, 0.05872165359954467966744972753, 0.40291360473737722197930117533, 0.58854728317271444063581659187, 1.38583430836009864903264407876, 1.59476994122708884207648717350, 1.85655848119167621908075159112, 1.90905473831419569390846587109, 1.99300006853509246466736611099, 2.82754981782417286067803685224, 2.99836881175627482680437891435, 3.42082938970506115434798711658, 3.54535807012569873530300333170, 3.77472306197706415176602451654, 4.12217912254924503675000079622, 4.26805590734628580262250456779, 4.73269767142320120667910626329, 4.92676933425425517985232750969, 5.32713873338853142455431870390, 5.41071178800638831500692382017, 5.75868635530745618459268134434, 6.07582074538987587300712214949, 6.23757582369929073334200856515, 6.34172043198974312739762736923, 6.96858691668564559204308352713, 7.00083443915422023261798862156

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