Properties

Label 8-4840e4-1.1-c1e4-0-1
Degree $8$
Conductor $5.488\times 10^{14}$
Sign $1$
Analytic cond. $2.23095\times 10^{6}$
Root an. cond. $6.21671$
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  − 2·3-s − 4·5-s − 6·7-s − 2·9-s − 4·13-s + 8·15-s + 12·21-s + 4·23-s + 10·25-s + 8·27-s − 16·29-s + 16·31-s + 24·35-s − 8·37-s + 8·39-s − 8·41-s + 10·43-s + 8·45-s + 14·47-s + 2·49-s + 8·53-s + 4·61-s + 12·63-s + 16·65-s − 2·67-s − 8·69-s + 8·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 2.26·7-s − 2/3·9-s − 1.10·13-s + 2.06·15-s + 2.61·21-s + 0.834·23-s + 2·25-s + 1.53·27-s − 2.97·29-s + 2.87·31-s + 4.05·35-s − 1.31·37-s + 1.28·39-s − 1.24·41-s + 1.52·43-s + 1.19·45-s + 2.04·47-s + 2/7·49-s + 1.09·53-s + 0.512·61-s + 1.51·63-s + 1.98·65-s − 0.244·67-s − 0.963·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.23095\times 10^{6}\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 2 p T^{2} + 8 T^{3} + 19 T^{4} + 8 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 34 T^{2} + 120 T^{3} + 375 T^{4} + 120 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 40 T^{2} + 124 T^{3} + 718 T^{4} + 124 p T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 40 T^{2} - 96 T^{3} + 750 T^{4} - 96 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 56 T^{2} - 292 T^{3} + 1534 T^{4} - 292 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 160 T^{2} - 1120 T^{3} + 6862 T^{4} - 1120 p T^{5} + 160 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 100 T^{2} + 632 T^{3} + 4918 T^{4} + 632 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 134 T^{2} + 992 T^{3} + 7627 T^{4} + 992 p T^{5} + 134 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 118 T^{2} - 784 T^{3} + 6571 T^{4} - 784 p T^{5} + 118 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 194 T^{2} - 1640 T^{3} + 13975 T^{4} - 1640 p T^{5} + 194 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 200 T^{2} - 1256 T^{3} + 15598 T^{4} - 1256 p T^{5} + 200 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 128 T^{2} - 432 T^{3} + 7710 T^{4} - 432 p T^{5} + 128 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 154 T^{2} - 832 T^{3} + 11575 T^{4} - 832 p T^{5} + 154 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 166 T^{2} + 632 T^{3} + 13843 T^{4} + 632 p T^{5} + 166 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 128 T^{2} - 1496 T^{3} + 11758 T^{4} - 1496 p T^{5} + 128 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 196 T^{2} - 292 T^{3} + 17734 T^{4} - 292 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 152 T^{2} + 924 T^{3} + 6798 T^{4} + 924 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 266 T^{2} - 1680 T^{3} + 27219 T^{4} - 1680 p T^{5} + 266 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 184 T^{2} + 528 T^{3} + 20286 T^{4} + 528 p T^{5} + 184 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.33438144126661484134299403308, −5.80049086187115752496346963005, −5.74485093274503838056724145792, −5.73513521410340656726552349723, −5.68223305892974714840753814264, −5.20065469472605375375174914658, −5.00085152878740162467658052885, −4.97813194788055542694192850371, −4.72103227892679813824362360749, −4.47196248710837285643259573384, −4.08569165877536759893768782276, −4.08199470780502729531894387540, −4.00748381671310289106426224997, −3.42346517721081404878611104844, −3.36043435530611953725181416520, −3.35222206698078201064879038639, −3.24894280062772603193755833185, −2.62537375799340254876179130037, −2.53115775905549530871287180072, −2.48555648855299335804546112440, −2.39359970725790053930261603716, −1.67195535143843282190004457515, −1.34166816743814632468553754744, −0.958875467130548123885609823546, −0.904865285434983717172145664640, 0, 0, 0, 0, 0.904865285434983717172145664640, 0.958875467130548123885609823546, 1.34166816743814632468553754744, 1.67195535143843282190004457515, 2.39359970725790053930261603716, 2.48555648855299335804546112440, 2.53115775905549530871287180072, 2.62537375799340254876179130037, 3.24894280062772603193755833185, 3.35222206698078201064879038639, 3.36043435530611953725181416520, 3.42346517721081404878611104844, 4.00748381671310289106426224997, 4.08199470780502729531894387540, 4.08569165877536759893768782276, 4.47196248710837285643259573384, 4.72103227892679813824362360749, 4.97813194788055542694192850371, 5.00085152878740162467658052885, 5.20065469472605375375174914658, 5.68223305892974714840753814264, 5.73513521410340656726552349723, 5.74485093274503838056724145792, 5.80049086187115752496346963005, 6.33438144126661484134299403308

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