Properties

Label 8-9800e4-1.1-c1e4-0-9
Degree $8$
Conductor $9.224\times 10^{15}$
Sign $1$
Analytic cond. $3.74983\times 10^{7}$
Root an. cond. $8.84609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 4·11-s − 4·23-s − 8·29-s + 8·37-s − 4·53-s − 12·67-s − 12·79-s + 14·81-s + 24·99-s − 28·109-s + 16·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·9-s − 1.20·11-s − 0.834·23-s − 1.48·29-s + 1.31·37-s − 0.549·53-s − 1.46·67-s − 1.35·79-s + 14/9·81-s + 2.41·99-s − 2.68·109-s + 1.50·113-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 − 3.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.74983\times 10^{7}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$D_4\times C_2$ \( 1 + 2 p T^{2} + 22 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 44 T^{2} + 982 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
29$D_{4}$ \( ( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 70 T^{2} + 2742 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 73 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 70 T^{2} + 2382 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 81 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 110 T^{2} + 6342 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 214 T^{2} + 18766 T^{4} + 214 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 97 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 118 T^{2} + 8014 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 262 T^{2} + 29814 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 326 T^{2} + 42286 T^{4} + 326 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 214 T^{2} + 24142 T^{4} + 214 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.90738106136691561757133579830, −5.42274501250610507180408713816, −5.35600076064509832459287828348, −5.30252674524540328930652526651, −5.15586505720173442557244190154, −4.79132723366823733200557038770, −4.70671298580679938848947888610, −4.51561140817822562080715525262, −4.31797664535834713360278932306, −3.98946923836800830449240857492, −3.85391315685408130592982566864, −3.69438448391207074587248076696, −3.66226818982007425179188156196, −3.04474631188254191884002749032, −3.02623203134196958893570091674, −3.01331907287267966781612295737, −2.87694476306456810369416400755, −2.32943961120081424099764942809, −2.29787459493158940355719275483, −2.24431956421536731978367102176, −2.04613641596007151669541670027, −1.55031827529014905477045036704, −1.29372091699976154238158452134, −1.03668577282323351620979187410, −0.974158296194249326665312033956, 0, 0, 0, 0, 0.974158296194249326665312033956, 1.03668577282323351620979187410, 1.29372091699976154238158452134, 1.55031827529014905477045036704, 2.04613641596007151669541670027, 2.24431956421536731978367102176, 2.29787459493158940355719275483, 2.32943961120081424099764942809, 2.87694476306456810369416400755, 3.01331907287267966781612295737, 3.02623203134196958893570091674, 3.04474631188254191884002749032, 3.66226818982007425179188156196, 3.69438448391207074587248076696, 3.85391315685408130592982566864, 3.98946923836800830449240857492, 4.31797664535834713360278932306, 4.51561140817822562080715525262, 4.70671298580679938848947888610, 4.79132723366823733200557038770, 5.15586505720173442557244190154, 5.30252674524540328930652526651, 5.35600076064509832459287828348, 5.42274501250610507180408713816, 5.90738106136691561757133579830

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