Properties

Label 8-126e4-1.1-c3e4-0-0
Degree $8$
Conductor $252047376$
Sign $1$
Analytic cond. $3054.54$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 4·4-s + 7·5-s + 6·7-s + 16·8-s − 28·10-s − 25·11-s − 98·13-s − 24·14-s − 64·16-s + 98·17-s + 119·19-s + 28·20-s + 100·22-s + 122·23-s + 214·25-s + 392·26-s + 24·28-s − 146·29-s − 98·31-s + 64·32-s − 392·34-s + 42·35-s − 289·37-s − 476·38-s + 112·40-s − 672·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.626·5-s + 0.323·7-s + 0.707·8-s − 0.885·10-s − 0.685·11-s − 2.09·13-s − 0.458·14-s − 16-s + 1.39·17-s + 1.43·19-s + 0.313·20-s + 0.969·22-s + 1.10·23-s + 1.71·25-s + 2.95·26-s + 0.161·28-s − 0.934·29-s − 0.567·31-s + 0.353·32-s − 1.97·34-s + 0.202·35-s − 1.28·37-s − 2.03·38-s + 0.442·40-s − 2.55·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3054.54\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.9061831421\)
\(L(\frac12)\) \(\approx\) \(0.9061831421\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 - 6 T - 11 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
good5$D_4\times C_2$ \( 1 - 7 T - 33 p T^{2} + 252 T^{3} + 24046 T^{4} + 252 p^{3} T^{5} - 33 p^{7} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 25 T + 171 T^{2} - 55200 T^{3} - 2397320 T^{4} - 55200 p^{3} T^{5} + 171 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 + 49 T + 3788 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 98 T + 2202 T^{2} + 237552 T^{3} - 16532417 T^{4} + 237552 p^{3} T^{5} + 2202 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 119 T - 2663 T^{2} - 369614 T^{3} + 138870796 T^{4} - 369614 p^{3} T^{5} - 2663 p^{6} T^{6} - 119 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 122 T - 3714 T^{2} + 699792 T^{3} + 16756087 T^{4} + 699792 p^{3} T^{5} - 3714 p^{6} T^{6} - 122 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 73 T + 47746 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 98 T - 14551 T^{2} - 3471846 T^{3} - 590152420 T^{4} - 3471846 p^{3} T^{5} - 14551 p^{6} T^{6} + 98 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 289 T - 17387 T^{2} - 115022 T^{3} + 3386108842 T^{4} - 115022 p^{3} T^{5} - 17387 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2$ \( ( 1 + 168 T + p^{3} T^{2} )^{4} \)
43$D_{4}$ \( ( 1 - 307 T + 161298 T^{2} - 307 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 672 T + 158834 T^{2} - 57189888 T^{3} + 28038541539 T^{4} - 57189888 p^{3} T^{5} + 158834 p^{6} T^{6} - 672 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 375 T - 781 T^{2} - 58630500 T^{3} - 23104532418 T^{4} - 58630500 p^{3} T^{5} - 781 p^{6} T^{6} + 375 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 763 T + 205407 T^{2} + 25938948 T^{3} - 19796216048 T^{4} + 25938948 p^{3} T^{5} + 205407 p^{6} T^{6} - 763 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 406 T - 274558 T^{2} + 5914608 T^{3} + 104132072759 T^{4} + 5914608 p^{3} T^{5} - 274558 p^{6} T^{6} - 406 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 1041 T + 270341 T^{2} - 220498374 T^{3} + 245132424828 T^{4} - 220498374 p^{3} T^{5} + 270341 p^{6} T^{6} - 1041 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 1652 T + 1360270 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 189 T - 297259 T^{2} + 84115206 T^{3} - 54354622146 T^{4} + 84115206 p^{3} T^{5} - 297259 p^{6} T^{6} - 189 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 524 T - 316753 T^{2} - 206848476 T^{3} - 28794145744 T^{4} - 206848476 p^{3} T^{5} - 316753 p^{6} T^{6} + 524 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 287 T + 781978 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 2394 T + 3029186 T^{2} - 3093316128 T^{3} + 2763749199855 T^{4} - 3093316128 p^{3} T^{5} + 3029186 p^{6} T^{6} - 2394 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 + 63 T + 667132 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−9.472243527677998301842396325131, −9.179868164251782194389409565392, −8.651864754241102813060213678523, −8.575193991709920056796504425158, −8.561596535214405508747673469796, −7.64929337415567143061603601223, −7.61696386326996933913805589159, −7.56892226174014047775280635041, −7.14514331579886921539529247145, −6.96796977731882550833358806452, −6.71975851433823225849749421875, −5.84087164261801601127020526136, −5.77833682129597488886901496070, −5.25083037808514774902100032873, −5.11852399797098339356760306197, −5.01973198956340014473638086608, −4.47221414244489050935324096215, −3.89980772426093100416428577611, −3.45178931008482918989633262652, −2.83641656124131941327720682150, −2.72022205158737906360544173768, −2.07297364917298969944364265602, −1.45850402007738676730336206853, −1.01532438566663772465764155964, −0.36641668518863912306773905752, 0.36641668518863912306773905752, 1.01532438566663772465764155964, 1.45850402007738676730336206853, 2.07297364917298969944364265602, 2.72022205158737906360544173768, 2.83641656124131941327720682150, 3.45178931008482918989633262652, 3.89980772426093100416428577611, 4.47221414244489050935324096215, 5.01973198956340014473638086608, 5.11852399797098339356760306197, 5.25083037808514774902100032873, 5.77833682129597488886901496070, 5.84087164261801601127020526136, 6.71975851433823225849749421875, 6.96796977731882550833358806452, 7.14514331579886921539529247145, 7.56892226174014047775280635041, 7.61696386326996933913805589159, 7.64929337415567143061603601223, 8.561596535214405508747673469796, 8.575193991709920056796504425158, 8.651864754241102813060213678523, 9.179868164251782194389409565392, 9.472243527677998301842396325131

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