Properties

Label 8-2800e4-1.1-c1e4-0-8
Degree $8$
Conductor $6.147\times 10^{13}$
Sign $1$
Analytic cond. $249885.$
Root an. cond. $4.72843$
Motivic weight $1$
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  + 3·9-s − 2·11-s − 12·19-s − 2·29-s + 4·41-s − 2·49-s − 16·59-s + 12·61-s − 32·71-s − 18·79-s − 7·81-s − 12·89-s − 6·99-s − 32·101-s − 46·109-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 31·169-s − 36·171-s + ⋯
L(s)  = 1  + 9-s − 0.603·11-s − 2.75·19-s − 0.371·29-s + 0.624·41-s − 2/7·49-s − 2.08·59-s + 1.53·61-s − 3.79·71-s − 2.02·79-s − 7/9·81-s − 1.27·89-s − 0.603·99-s − 3.18·101-s − 4.40·109-s − 3·121-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 + 2.38·169-s − 2.75·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(249885.\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.422747287\)
\(L(\frac12)\) \(\approx\) \(1.422747287\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 47 T^{2} + 1024 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 176 T^{2} + 13294 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 124 T^{2} + 11734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 135 T^{2} + 14768 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.23078794631190053797573737869, −5.99801781294293740747761391398, −5.97889204769977280278178225125, −5.55622158556878941538398932387, −5.39636345351657888057026933719, −5.22072200494949401760437281947, −5.16225057799263856675908291233, −4.65738583788876981444789915797, −4.40014496932695222768942383595, −4.34551266295589190708727160942, −4.30388525667918335338218214118, −3.87603829961958412773186057119, −3.82919342115948216994089682906, −3.75713174986157639178994331638, −2.98029484433187708186807140340, −2.92953428440271108618337284769, −2.68528846344115209799596523772, −2.64221845860577361641568807484, −2.38940677915063738726660627051, −1.67465250342212139032783354021, −1.66556931333846498129795453625, −1.48428517972698366486251587448, −1.35076573566432879273001582546, −0.39792314554983039333910740301, −0.30399013352258424053596348510, 0.30399013352258424053596348510, 0.39792314554983039333910740301, 1.35076573566432879273001582546, 1.48428517972698366486251587448, 1.66556931333846498129795453625, 1.67465250342212139032783354021, 2.38940677915063738726660627051, 2.64221845860577361641568807484, 2.68528846344115209799596523772, 2.92953428440271108618337284769, 2.98029484433187708186807140340, 3.75713174986157639178994331638, 3.82919342115948216994089682906, 3.87603829961958412773186057119, 4.30388525667918335338218214118, 4.34551266295589190708727160942, 4.40014496932695222768942383595, 4.65738583788876981444789915797, 5.16225057799263856675908291233, 5.22072200494949401760437281947, 5.39636345351657888057026933719, 5.55622158556878941538398932387, 5.97889204769977280278178225125, 5.99801781294293740747761391398, 6.23078794631190053797573737869

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