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Properties

Label 8-3675e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.824\times 10^{14}$
Sign $1$
Analytic cond. $741545.$
Root an. cond. $5.41710$
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  + 2·2-s + 4·3-s − 4-s + 8·6-s − 6·8-s + 10·9-s − 4·11-s − 4·12-s − 6·16-s − 4·17-s + 20·18-s + 8·19-s − 8·22-s − 24·24-s + 20·27-s − 4·29-s + 8·31-s + 2·32-s − 16·33-s − 8·34-s − 10·36-s + 16·37-s + 16·38-s + 24·41-s + 20·43-s + 4·44-s + 8·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s − 1/2·4-s + 3.26·6-s − 2.12·8-s + 10/3·9-s − 1.20·11-s − 1.15·12-s − 3/2·16-s − 0.970·17-s + 4.71·18-s + 1.83·19-s − 1.70·22-s − 4.89·24-s + 3.84·27-s − 0.742·29-s + 1.43·31-s + 0.353·32-s − 2.78·33-s − 1.37·34-s − 5/3·36-s + 2.63·37-s + 2.59·38-s + 3.74·41-s + 3.04·43-s + 0.603·44-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(741545.\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(29.53247804\)
\(L(\frac12)\) \(\approx\) \(29.53247804\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
7 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + 5 T^{2} - 3 p T^{3} + 11 T^{4} - 3 p^{2} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 36 T^{2} + 124 T^{3} + 557 T^{4} + 124 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 38 T^{2} - 16 T^{3} + 654 T^{4} - 16 p T^{5} + 38 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 58 T^{2} + 188 T^{3} + 1390 T^{4} + 188 p T^{5} + 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 46 T^{2} - 232 T^{3} + 1222 T^{4} - 232 p T^{5} + 46 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 60 T^{2} - 20 T^{3} + 1781 T^{4} - 20 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 68 T^{2} + 284 T^{3} + 2725 T^{4} + 284 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 88 T^{2} - 600 T^{3} + 3986 T^{4} - 600 p T^{5} + 88 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 194 T^{2} - 1584 T^{3} + 10995 T^{4} - 1584 p T^{5} + 194 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 348 T^{2} - 3464 T^{3} + 25622 T^{4} - 3464 p T^{5} + 348 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 170 T^{2} - 600 T^{3} + 1347 T^{4} - 600 p T^{5} + 170 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 86 T^{2} - 224 T^{3} + 2638 T^{4} - 224 p T^{5} + 86 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 304 T^{2} + 3100 T^{3} + 25822 T^{4} + 3100 p T^{5} + 304 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 158 T^{2} - 1168 T^{3} + 11998 T^{4} - 1168 p T^{5} + 158 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 194 T^{2} - 2008 T^{3} + 18507 T^{4} - 2008 p T^{5} + 194 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 116 T^{2} - 348 T^{3} + 9773 T^{4} - 348 p T^{5} + 116 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 134 T^{2} + 160 T^{3} + 11790 T^{4} + 160 p T^{5} + 134 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 + 114 T^{2} + 7539 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 292 T^{2} + 3004 T^{3} + 28002 T^{4} + 3004 p T^{5} + 292 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 142 T^{2} + 368 T^{3} + 2742 T^{4} + 368 p T^{5} + 142 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 496 T^{2} - 6296 T^{3} + 73730 T^{4} - 6296 p T^{5} + 496 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.04796352998488076766433378357, −5.56078225387290423212885145438, −5.53280475255673919275932642693, −5.45076611367204720915121020877, −5.31336874017925897545188455870, −4.69777776693088268628622376539, −4.62763210195498722436269772743, −4.60126474367195640489828063718, −4.53395818137399618287663327071, −4.04366113045509980520665971127, −3.98220589931544690382308814443, −3.85450084459741174544232426409, −3.80571678389524094881409955202, −3.31316575394926089368797259388, −3.08348747801872720882390474239, −2.91511898301884970319532335010, −2.66140526393656216456557018105, −2.43755481202205707760238074556, −2.40020464711456605700646619645, −2.27915860417352155759525098868, −1.88394235222843770020237138488, −1.27634698237330861938207667494, −0.927833404789528854069126535125, −0.67698962851645896417576382843, −0.63870405430360973043362896765, 0.63870405430360973043362896765, 0.67698962851645896417576382843, 0.927833404789528854069126535125, 1.27634698237330861938207667494, 1.88394235222843770020237138488, 2.27915860417352155759525098868, 2.40020464711456605700646619645, 2.43755481202205707760238074556, 2.66140526393656216456557018105, 2.91511898301884970319532335010, 3.08348747801872720882390474239, 3.31316575394926089368797259388, 3.80571678389524094881409955202, 3.85450084459741174544232426409, 3.98220589931544690382308814443, 4.04366113045509980520665971127, 4.53395818137399618287663327071, 4.60126474367195640489828063718, 4.62763210195498722436269772743, 4.69777776693088268628622376539, 5.31336874017925897545188455870, 5.45076611367204720915121020877, 5.53280475255673919275932642693, 5.56078225387290423212885145438, 6.04796352998488076766433378357

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