Properties

Label 8-147e4-1.1-c3e4-0-3
Degree $8$
Conductor $466948881$
Sign $1$
Analytic cond. $5658.92$
Root an. cond. $2.94504$
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  + 3·2-s + 6·3-s + 4·4-s + 6·5-s + 18·6-s − 39·8-s + 9·9-s + 18·10-s + 6·11-s + 24·12-s − 32·13-s + 36·15-s − 125·16-s − 6·17-s + 27·18-s + 64·19-s + 24·20-s + 18·22-s − 6·23-s − 234·24-s + 202·25-s − 96·26-s − 54·27-s − 504·29-s + 108·30-s + 40·31-s − 252·32-s + ⋯
L(s)  = 1  + 1.06·2-s + 1.15·3-s + 1/2·4-s + 0.536·5-s + 1.22·6-s − 1.72·8-s + 1/3·9-s + 0.569·10-s + 0.164·11-s + 0.577·12-s − 0.682·13-s + 0.619·15-s − 1.95·16-s − 0.0856·17-s + 0.353·18-s + 0.772·19-s + 0.268·20-s + 0.174·22-s − 0.0543·23-s − 1.99·24-s + 1.61·25-s − 0.724·26-s − 0.384·27-s − 3.22·29-s + 0.657·30-s + 0.231·31-s − 1.39·32-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(5658.92\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.096065592\)
\(L(\frac12)\) \(\approx\) \(8.096065592\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 - 3 T + 5 T^{2} + 9 p^{2} T^{3} - 15 p^{3} T^{4} + 9 p^{5} T^{5} + 5 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
5$D_4\times C_2$ \( 1 - 6 T - 166 T^{2} + 288 T^{3} + 20679 T^{4} + 288 p^{3} T^{5} - 166 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 6 T - 10 p^{2} T^{2} + 8496 T^{3} - 266961 T^{4} + 8496 p^{3} T^{5} - 10 p^{8} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 6 T - 9742 T^{2} - 288 T^{3} + 71294847 T^{4} - 288 p^{3} T^{5} - 9742 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 64 T - 2438 T^{2} + 459776 T^{3} - 32447189 T^{4} + 459776 p^{3} T^{5} - 2438 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 6 T - 7834 T^{2} - 98784 T^{3} - 86537001 T^{4} - 98784 p^{3} T^{5} - 7834 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 40 T + 15490 T^{2} + 2938880 T^{3} - 742237181 T^{4} + 2938880 p^{3} T^{5} + 15490 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 248 T - 36710 T^{2} + 766816 T^{3} + 3964901275 T^{4} + 766816 p^{3} T^{5} - 36710 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 450 T + 175642 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 12 T - 141646 T^{2} - 790272 T^{3} + 9310238259 T^{4} - 790272 p^{3} T^{5} - 141646 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 1104 T + 616586 T^{2} - 336141504 T^{3} + 159062942139 T^{4} - 336141504 p^{3} T^{5} + 616586 p^{6} T^{6} - 1104 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 804 T + 265802 T^{2} + 24235776 T^{3} - 30073788309 T^{4} + 24235776 p^{3} T^{5} + 265802 p^{6} T^{6} - 804 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 428 T - 242702 T^{2} - 12016528 T^{3} + 88279223131 T^{4} - 12016528 p^{3} T^{5} - 242702 p^{6} T^{6} + 428 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 148 T - 418886 T^{2} - 23788928 T^{3} + 97249529179 T^{4} - 23788928 p^{3} T^{5} - 418886 p^{6} T^{6} + 148 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1072 T + 85906 T^{2} - 305781568 T^{3} + 532173766867 T^{4} - 305781568 p^{3} T^{5} + 85906 p^{6} T^{6} - 1072 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 - 572 T - 574478 T^{2} + 48285952 T^{3} + 408592434547 T^{4} + 48285952 p^{3} T^{5} - 574478 p^{6} T^{6} - 572 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 1944 T + 1957030 T^{2} + 1944 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 366 T - 1022134 T^{2} + 92908368 T^{3} + 745127969775 T^{4} + 92908368 p^{3} T^{5} - 1022134 p^{6} T^{6} - 366 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−9.164669312560782504395326677539, −9.043373294422629179109165813813, −8.546031433195197709454924987401, −8.367227899367683212636453616432, −8.021961406744231743930246604127, −7.47982887724004432717000564509, −7.40380604244059676606981974904, −7.26596724723005917917315435989, −6.65100765796960519966417657995, −6.59551480143582557645161233892, −5.94096839688211398652261785144, −5.71328713187058663646586476036, −5.58613245136338249929406973264, −5.45309706896929919301865269848, −5.00582069417998540101723612783, −4.27894490994788211800145719862, −4.03057493186719304582028083233, −3.89248590536179903575297532426, −3.50716187894208097559744738590, −2.76177925592101989715167850422, −2.63698755685416889172051284077, −2.58345253052760990391276098169, −2.02271058530053010935297252676, −1.04052776790434379072812831781, −0.54026771223422295548329415948, 0.54026771223422295548329415948, 1.04052776790434379072812831781, 2.02271058530053010935297252676, 2.58345253052760990391276098169, 2.63698755685416889172051284077, 2.76177925592101989715167850422, 3.50716187894208097559744738590, 3.89248590536179903575297532426, 4.03057493186719304582028083233, 4.27894490994788211800145719862, 5.00582069417998540101723612783, 5.45309706896929919301865269848, 5.58613245136338249929406973264, 5.71328713187058663646586476036, 5.94096839688211398652261785144, 6.59551480143582557645161233892, 6.65100765796960519966417657995, 7.26596724723005917917315435989, 7.40380604244059676606981974904, 7.47982887724004432717000564509, 8.021961406744231743930246604127, 8.367227899367683212636453616432, 8.546031433195197709454924987401, 9.043373294422629179109165813813, 9.164669312560782504395326677539

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