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Properties

Label 8-42e8-1.1-c3e4-0-13
Degree $8$
Conductor $9.683\times 10^{12}$
Sign $1$
Analytic cond. $1.17343\times 10^{8}$
Root an. cond. $10.2019$
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  + 48·17-s − 192·19-s − 192·23-s − 88·25-s − 96·29-s − 48·31-s + 256·37-s + 1.00e3·41-s − 112·43-s + 864·47-s + 648·53-s + 336·59-s − 960·61-s + 720·67-s + 1.34e3·71-s − 672·73-s − 1.98e3·79-s + 3.12e3·83-s + 2.16e3·89-s − 2.01e3·97-s + 192·101-s − 2.54e3·103-s + 1.82e3·107-s + 688·109-s + 1.03e3·113-s − 1.18e3·121-s + 576·125-s + ⋯
L(s)  = 1  + 0.684·17-s − 2.31·19-s − 1.74·23-s − 0.703·25-s − 0.614·29-s − 0.278·31-s + 1.13·37-s + 3.83·41-s − 0.397·43-s + 2.68·47-s + 1.67·53-s + 0.741·59-s − 2.01·61-s + 1.31·67-s + 2.24·71-s − 1.07·73-s − 2.82·79-s + 4.12·83-s + 2.57·89-s − 2.11·97-s + 0.189·101-s − 2.43·103-s + 1.64·107-s + 0.604·109-s + 0.859·113-s − 0.892·121-s + 0.412·125-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.17343\times 10^{8}\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.081928032\)
\(L(\frac12)\) \(\approx\) \(8.081928032\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 88 T^{2} - 576 T^{3} + 18498 T^{4} - 576 p^{3} T^{5} + 88 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 108 p T^{2} - 82944 T^{3} + 352406 T^{4} - 82944 p^{3} T^{5} + 108 p^{7} T^{6} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 1696 T^{2} + 31104 T^{3} + 7806642 T^{4} + 31104 p^{3} T^{5} + 1696 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 9880 T^{2} - 116976 T^{3} + 43940658 T^{4} - 116976 p^{3} T^{5} + 9880 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 192 T + 24156 T^{2} + 2400960 T^{3} + 195309110 T^{4} + 2400960 p^{3} T^{5} + 24156 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 192 T + 52596 T^{2} + 6099648 T^{3} + 943244678 T^{4} + 6099648 p^{3} T^{5} + 52596 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 96 T + 85860 T^{2} + 6375072 T^{3} + 3037151990 T^{4} + 6375072 p^{3} T^{5} + 85860 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 75532 T^{2} - 1415952 T^{3} + 2535445158 T^{4} - 1415952 p^{3} T^{5} + 75532 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 256 T + 66772 T^{2} + 2259200 T^{3} + 153679222 T^{4} + 2259200 p^{3} T^{5} + 66772 p^{6} T^{6} - 256 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 1008 T + 609784 T^{2} - 250033968 T^{3} + 76165467474 T^{4} - 250033968 p^{3} T^{5} + 609784 p^{6} T^{6} - 1008 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 112 T + 246988 T^{2} + 11722480 T^{3} + 25842449782 T^{4} + 11722480 p^{3} T^{5} + 246988 p^{6} T^{6} + 112 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 864 T + 642220 T^{2} - 282465504 T^{3} + 110627505318 T^{4} - 282465504 p^{3} T^{5} + 642220 p^{6} T^{6} - 864 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 648 T + 386892 T^{2} - 114656472 T^{3} + 50365144694 T^{4} - 114656472 p^{3} T^{5} + 386892 p^{6} T^{6} - 648 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 336 T + 534364 T^{2} - 159356496 T^{3} + 154119406422 T^{4} - 159356496 p^{3} T^{5} + 534364 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 960 T + 577728 T^{2} + 153988800 T^{3} + 52569538418 T^{4} + 153988800 p^{3} T^{5} + 577728 p^{6} T^{6} + 960 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 720 T + 533740 T^{2} - 318964176 T^{3} + 154478344470 T^{4} - 318964176 p^{3} T^{5} + 533740 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1344 T + 1892084 T^{2} - 1427151168 T^{3} + 1080205217862 T^{4} - 1427151168 p^{3} T^{5} + 1892084 p^{6} T^{6} - 1344 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 672 T + 1088640 T^{2} + 325175712 T^{3} + 453343664738 T^{4} + 325175712 p^{3} T^{5} + 1088640 p^{6} T^{6} + 672 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1984 T + 1725436 T^{2} + 666213568 T^{3} + 202189693510 T^{4} + 666213568 p^{3} T^{5} + 1725436 p^{6} T^{6} + 1984 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3120 T + 5531020 T^{2} - 6555475248 T^{3} + 5745505983510 T^{4} - 6555475248 p^{3} T^{5} + 5531020 p^{6} T^{6} - 3120 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 2160 T + 3343000 T^{2} - 3573514800 T^{3} + 3407992760850 T^{4} - 3573514800 p^{3} T^{5} + 3343000 p^{6} T^{6} - 2160 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2016 T + 2898432 T^{2} + 2840274144 T^{3} + 3044116636418 T^{4} + 2840274144 p^{3} T^{5} + 2898432 p^{6} T^{6} + 2016 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.31223469239438908155158644968, −5.98709077833065777667266468589, −5.71311126275530342596819753687, −5.69560608933825795733586324940, −5.59720813655768288505918503432, −5.16359622403488048569727342185, −5.06292074714001884499185915277, −4.51447467953315895157740228742, −4.40929974488391429139251651484, −4.21182511384713283381241760971, −3.95400239308725660352877293616, −3.93085655362888110026282159963, −3.88290683341506184642268350630, −3.33795175033010433243333321443, −2.89096471932008851650028836790, −2.73387827838392153395956150467, −2.70652283793854025036179082952, −2.05110625121299671118286703014, −2.00959072363156848984600150792, −1.96263909827835705532463874887, −1.66128963728593944314225337605, −0.875162236366067393268783393992, −0.66788402554379578868326489412, −0.66116521202031232703735183071, −0.36493675076773575246073066020, 0.36493675076773575246073066020, 0.66116521202031232703735183071, 0.66788402554379578868326489412, 0.875162236366067393268783393992, 1.66128963728593944314225337605, 1.96263909827835705532463874887, 2.00959072363156848984600150792, 2.05110625121299671118286703014, 2.70652283793854025036179082952, 2.73387827838392153395956150467, 2.89096471932008851650028836790, 3.33795175033010433243333321443, 3.88290683341506184642268350630, 3.93085655362888110026282159963, 3.95400239308725660352877293616, 4.21182511384713283381241760971, 4.40929974488391429139251651484, 4.51447467953315895157740228742, 5.06292074714001884499185915277, 5.16359622403488048569727342185, 5.59720813655768288505918503432, 5.69560608933825795733586324940, 5.71311126275530342596819753687, 5.98709077833065777667266468589, 6.31223469239438908155158644968

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