Properties

Label 8-3e16-1.1-c8e4-0-3
Degree $8$
Conductor $43046721$
Sign $1$
Analytic cond. $1.18558\times 10^{6}$
Root an. cond. $5.74435$
Motivic weight $8$
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  − 494·4-s − 3.30e3·7-s + 5.25e4·13-s + 6.55e4·16-s + 1.86e5·19-s + 1.95e5·25-s + 1.63e6·28-s + 3.92e5·31-s + 1.12e7·37-s + 4.42e6·43-s + 1.42e7·49-s − 2.59e7·52-s + 3.48e7·61-s + 2.34e7·64-s + 2.86e7·67-s − 3.56e7·73-s − 9.21e7·76-s − 6.55e7·79-s − 1.73e8·91-s + 4.89e7·97-s − 9.68e7·100-s − 3.28e8·103-s − 2.97e8·109-s − 2.16e8·112-s + 2.77e7·121-s − 1.94e8·124-s + 127-s + ⋯
L(s)  = 1  − 1.92·4-s − 1.37·7-s + 1.83·13-s + 16-s + 1.43·19-s + 0.501·25-s + 2.65·28-s + 0.425·31-s + 6.01·37-s + 1.29·43-s + 2.47·49-s − 3.55·52-s + 2.51·61-s + 1.39·64-s + 1.42·67-s − 1.25·73-s − 2.76·76-s − 1.68·79-s − 2.53·91-s + 0.552·97-s − 0.968·100-s − 2.91·103-s − 2.10·109-s − 1.37·112-s + 0.129·121-s − 0.820·124-s − 1.96·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(43046721\)    =    \(3^{16}\)
Sign: $1$
Analytic conductor: \(1.18558\times 10^{6}\)
Root analytic conductor: \(5.74435\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 43046721,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.231933365\)
\(L(\frac12)\) \(\approx\) \(4.231933365\)
\(L(5)\) 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 \( 1 \)
good2$C_2^3$ \( 1 + 247 p T^{2} + 44625 p^{2} T^{4} + 247 p^{17} T^{6} + p^{32} T^{8} \)
5$C_2^3$ \( 1 - 195952 T^{2} - 114190704321 T^{4} - 195952 p^{16} T^{6} + p^{32} T^{8} \)
7$C_2^2$ \( ( 1 + 236 p T - 61953 p^{2} T^{2} + 236 p^{9} T^{3} + p^{16} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 27785566 T^{2} - 45177692185631805 T^{4} - 27785566 p^{16} T^{6} + p^{32} T^{8} \)
13$C_2^2$ \( ( 1 - 26272 T - 125512737 T^{2} - 26272 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 13720948544 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 46640 T + p^{8} T^{2} )^{4} \)
23$C_2^3$ \( 1 + 48977682530 T^{2} - \)\(37\!\cdots\!61\)\( T^{4} + 48977682530 p^{16} T^{6} + p^{32} T^{8} \)
29$C_2^3$ \( 1 + 622288126160 T^{2} + \)\(13\!\cdots\!79\)\( T^{4} + 622288126160 p^{16} T^{6} + p^{32} T^{8} \)
31$C_2^2$ \( ( 1 - 196444 T - 814300792305 T^{2} - 196444 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 2819414 T + p^{8} T^{2} )^{4} \)
41$C_2^3$ \( 1 + 15470807999360 T^{2} + \)\(17\!\cdots\!59\)\( T^{4} + 15470807999360 p^{16} T^{6} + p^{32} T^{8} \)
43$C_2^2$ \( ( 1 - 2213464 T - 6788777398305 T^{2} - 2213464 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 44956019993570 T^{2} + \)\(14\!\cdots\!79\)\( T^{4} + 44956019993570 p^{16} T^{6} + p^{32} T^{8} \)
53$C_2^2$ \( ( 1 - 97597007293520 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 155721734882114 T^{2} + \)\(26\!\cdots\!55\)\( T^{4} + 155721734882114 p^{16} T^{6} + p^{32} T^{8} \)
61$C_2^2$ \( ( 1 - 17405302 T + 111237224713923 T^{2} - 17405302 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 14322664 T - 200928973499745 T^{2} - 14322664 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 1046721637212770 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 8906992 T + p^{8} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 32758844 T - 443966949690225 T^{2} + 32758844 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 2735523396260830 T^{2} + \)\(24\!\cdots\!19\)\( T^{4} - 2735523396260830 p^{16} T^{6} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 - 4399714222730624 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 24451744 T - 7239545809735425 T^{2} - 24451744 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−9.196762209164371645541484894787, −8.718502612867109998857955730414, −8.266335897152443736562700274815, −8.085484566185545131402728347474, −7.922280412604441713269169551653, −7.51064223252407096260812069789, −6.83726084465318284080804994524, −6.78362916068316930614551735083, −6.45590470748317321375320401567, −5.94600093692628505502799188119, −5.64911988086676347427566658780, −5.54258148342850752718586295885, −5.15409976186829364691440855033, −4.38449421487908619135508711667, −4.14053708880363459046697080103, −4.13307933752046929010793299540, −3.96371565046574526118685640357, −3.08174578579570626143193285638, −2.80777566647148489015012391158, −2.77764603681546226150170549435, −1.99349239952987206428463501729, −1.17300252762117206747338571810, −0.855206382081912791906581906256, −0.62814441521716116882981718992, −0.52220379549990212006971724388, 0.52220379549990212006971724388, 0.62814441521716116882981718992, 0.855206382081912791906581906256, 1.17300252762117206747338571810, 1.99349239952987206428463501729, 2.77764603681546226150170549435, 2.80777566647148489015012391158, 3.08174578579570626143193285638, 3.96371565046574526118685640357, 4.13307933752046929010793299540, 4.14053708880363459046697080103, 4.38449421487908619135508711667, 5.15409976186829364691440855033, 5.54258148342850752718586295885, 5.64911988086676347427566658780, 5.94600093692628505502799188119, 6.45590470748317321375320401567, 6.78362916068316930614551735083, 6.83726084465318284080804994524, 7.51064223252407096260812069789, 7.922280412604441713269169551653, 8.085484566185545131402728347474, 8.266335897152443736562700274815, 8.718502612867109998857955730414, 9.196762209164371645541484894787

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