Properties

Degree 8
Conductor $ 3^{4} $
Sign $1$
Motivic weight 41
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.98e4·2-s + 1.39e10·3-s − 1.71e12·4-s + 1.18e14·5-s − 9.73e14·6-s + 1.50e17·7-s + 2.26e18·8-s + 1.21e20·9-s − 8.27e18·10-s + 7.24e20·11-s − 2.39e22·12-s − 8.84e21·13-s − 1.04e22·14-s + 1.65e24·15-s − 3.10e24·16-s − 3.82e25·17-s − 8.48e24·18-s + 2.61e26·19-s − 2.03e26·20-s + 2.09e27·21-s − 5.06e25·22-s − 1.53e28·23-s + 3.15e28·24-s − 2.52e28·25-s + 6.17e26·26-s + 8.47e29·27-s − 2.58e29·28-s + ⋯
L(s)  = 1  − 0.0470·2-s + 2.30·3-s − 0.781·4-s + 0.555·5-s − 0.108·6-s + 0.711·7-s + 0.694·8-s + 10/3·9-s − 0.0261·10-s + 0.324·11-s − 1.80·12-s − 0.129·13-s − 0.0335·14-s + 1.28·15-s − 0.642·16-s − 2.28·17-s − 0.156·18-s + 1.59·19-s − 0.434·20-s + 1.64·21-s − 0.0152·22-s − 1.87·23-s + 1.60·24-s − 0.555·25-s + 0.00607·26-s + 3.84·27-s − 0.556·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+41/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  induced by $\chi_{3} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 81,\ (\ :41/2, 41/2, 41/2, 41/2),\ 1)\)
\(L(21)\)  \(\approx\)  \(6.008481472\)
\(L(\frac12)\)  \(\approx\)  \(6.008481472\)
\(L(\frac{43}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 8. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{20} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 34911 p T + 53875321015 p^{5} T^{2} - 247085990567271 p^{13} T^{3} + 171951681772402383 p^{25} T^{4} - 247085990567271 p^{54} T^{5} + 53875321015 p^{87} T^{6} + 34911 p^{124} T^{7} + p^{164} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 23707392755256 p T + \)\(31\!\cdots\!68\)\( p^{3} T^{2} - \)\(16\!\cdots\!68\)\( p^{7} T^{3} + \)\(21\!\cdots\!46\)\( p^{13} T^{4} - \)\(16\!\cdots\!68\)\( p^{48} T^{5} + \)\(31\!\cdots\!68\)\( p^{85} T^{6} - 23707392755256 p^{124} T^{7} + p^{164} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 150256264888927136 T + \)\(23\!\cdots\!48\)\( p^{2} T^{2} - \)\(55\!\cdots\!56\)\( p^{4} T^{3} + \)\(17\!\cdots\!90\)\( p^{9} T^{4} - \)\(55\!\cdots\!56\)\( p^{45} T^{5} + \)\(23\!\cdots\!48\)\( p^{84} T^{6} - 150256264888927136 p^{123} T^{7} + p^{164} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 65891877021040062096 p T + \)\(18\!\cdots\!32\)\( p^{2} T^{2} - \)\(63\!\cdots\!80\)\( p^{5} T^{3} + \)\(16\!\cdots\!46\)\( p^{8} T^{4} - \)\(63\!\cdots\!80\)\( p^{46} T^{5} + \)\(18\!\cdots\!32\)\( p^{84} T^{6} - 65891877021040062096 p^{124} T^{7} + p^{164} T^{8} \)
13$C_2 \wr S_4$ \( 1 + \)\(88\!\cdots\!08\)\( T + \)\(65\!\cdots\!88\)\( p T^{2} - \)\(38\!\cdots\!72\)\( p^{3} T^{3} + \)\(12\!\cdots\!50\)\( p^{5} T^{4} - \)\(38\!\cdots\!72\)\( p^{44} T^{5} + \)\(65\!\cdots\!88\)\( p^{83} T^{6} + \)\(88\!\cdots\!08\)\( p^{123} T^{7} + p^{164} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(38\!\cdots\!88\)\( T + \)\(68\!\cdots\!16\)\( p T^{2} + \)\(48\!\cdots\!60\)\( p^{3} T^{3} + \)\(33\!\cdots\!18\)\( p^{5} T^{4} + \)\(48\!\cdots\!60\)\( p^{44} T^{5} + \)\(68\!\cdots\!16\)\( p^{83} T^{6} + \)\(38\!\cdots\!88\)\( p^{123} T^{7} + p^{164} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(26\!\cdots\!48\)\( T + \)\(39\!\cdots\!48\)\( p T^{2} - \)\(21\!\cdots\!64\)\( p^{3} T^{3} + \)\(10\!\cdots\!66\)\( p^{5} T^{4} - \)\(21\!\cdots\!64\)\( p^{44} T^{5} + \)\(39\!\cdots\!48\)\( p^{83} T^{6} - \)\(26\!\cdots\!48\)\( p^{123} T^{7} + p^{164} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(15\!\cdots\!32\)\( T + \)\(86\!\cdots\!76\)\( p T^{2} + \)\(28\!\cdots\!52\)\( p^{2} T^{3} + \)\(10\!\cdots\!50\)\( p^{3} T^{4} + \)\(28\!\cdots\!52\)\( p^{43} T^{5} + \)\(86\!\cdots\!76\)\( p^{83} T^{6} + \)\(15\!\cdots\!32\)\( p^{123} T^{7} + p^{164} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!64\)\( T + \)\(94\!\cdots\!40\)\( p T^{2} + \)\(31\!\cdots\!88\)\( p^{2} T^{3} + \)\(13\!\cdots\!82\)\( p^{3} T^{4} + \)\(31\!\cdots\!88\)\( p^{43} T^{5} + \)\(94\!\cdots\!40\)\( p^{83} T^{6} + \)\(10\!\cdots\!64\)\( p^{123} T^{7} + p^{164} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(92\!\cdots\!04\)\( T + \)\(25\!\cdots\!88\)\( p T^{2} - \)\(41\!\cdots\!52\)\( p^{2} T^{3} + \)\(59\!\cdots\!54\)\( p^{3} T^{4} - \)\(41\!\cdots\!52\)\( p^{43} T^{5} + \)\(25\!\cdots\!88\)\( p^{83} T^{6} - \)\(92\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(14\!\cdots\!16\)\( p T^{2} - \)\(62\!\cdots\!84\)\( p^{2} T^{3} + \)\(31\!\cdots\!90\)\( p^{3} T^{4} - \)\(62\!\cdots\!84\)\( p^{43} T^{5} + \)\(14\!\cdots\!16\)\( p^{83} T^{6} - \)\(20\!\cdots\!56\)\( p^{123} T^{7} + p^{164} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(23\!\cdots\!04\)\( T + \)\(12\!\cdots\!88\)\( p T^{2} - \)\(70\!\cdots\!72\)\( T^{3} + \)\(99\!\cdots\!94\)\( T^{4} - \)\(70\!\cdots\!72\)\( p^{41} T^{5} + \)\(12\!\cdots\!88\)\( p^{83} T^{6} - \)\(23\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \)
43$C_2 \wr S_4$ \( 1 - \)\(39\!\cdots\!60\)\( T + \)\(29\!\cdots\!00\)\( T^{2} - \)\(85\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!98\)\( T^{4} - \)\(85\!\cdots\!80\)\( p^{41} T^{5} + \)\(29\!\cdots\!00\)\( p^{82} T^{6} - \)\(39\!\cdots\!60\)\( p^{123} T^{7} + p^{164} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(88\!\cdots\!20\)\( T + \)\(10\!\cdots\!20\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!18\)\( T^{4} + \)\(42\!\cdots\!40\)\( p^{41} T^{5} + \)\(10\!\cdots\!20\)\( p^{82} T^{6} + \)\(88\!\cdots\!20\)\( p^{123} T^{7} + p^{164} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(95\!\cdots\!28\)\( T + \)\(13\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(85\!\cdots\!10\)\( T^{4} - \)\(10\!\cdots\!72\)\( p^{41} T^{5} + \)\(13\!\cdots\!88\)\( p^{82} T^{6} - \)\(95\!\cdots\!28\)\( p^{123} T^{7} + p^{164} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!08\)\( T + \)\(66\!\cdots\!52\)\( T^{2} + \)\(24\!\cdots\!16\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!16\)\( p^{41} T^{5} + \)\(66\!\cdots\!52\)\( p^{82} T^{6} + \)\(18\!\cdots\!08\)\( p^{123} T^{7} + p^{164} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(53\!\cdots\!40\)\( T + \)\(67\!\cdots\!56\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(24\!\cdots\!60\)\( p^{41} T^{5} + \)\(67\!\cdots\!56\)\( p^{82} T^{6} - \)\(53\!\cdots\!40\)\( p^{123} T^{7} + p^{164} T^{8} \)
67$C_2 \wr S_4$ \( 1 + \)\(73\!\cdots\!28\)\( T + \)\(45\!\cdots\!12\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!06\)\( T^{4} + \)\(17\!\cdots\!00\)\( p^{41} T^{5} + \)\(45\!\cdots\!12\)\( p^{82} T^{6} + \)\(73\!\cdots\!28\)\( p^{123} T^{7} + p^{164} T^{8} \)
71$C_2 \wr S_4$ \( 1 + \)\(84\!\cdots\!52\)\( T + \)\(22\!\cdots\!48\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(13\!\cdots\!64\)\( p^{41} T^{5} + \)\(22\!\cdots\!48\)\( p^{82} T^{6} + \)\(84\!\cdots\!52\)\( p^{123} T^{7} + p^{164} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(44\!\cdots\!32\)\( T + \)\(51\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!72\)\( p^{41} T^{5} + \)\(51\!\cdots\!68\)\( p^{82} T^{6} + \)\(44\!\cdots\!32\)\( p^{123} T^{7} + p^{164} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(22\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!46\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{41} T^{5} + \)\(22\!\cdots\!16\)\( p^{82} T^{6} - \)\(14\!\cdots\!80\)\( p^{123} T^{7} + p^{164} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(15\!\cdots\!04\)\( T + \)\(99\!\cdots\!00\)\( T^{2} - \)\(72\!\cdots\!24\)\( T^{3} + \)\(39\!\cdots\!46\)\( T^{4} - \)\(72\!\cdots\!24\)\( p^{41} T^{5} + \)\(99\!\cdots\!00\)\( p^{82} T^{6} - \)\(15\!\cdots\!04\)\( p^{123} T^{7} + p^{164} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(39\!\cdots\!72\)\( T + \)\(90\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!46\)\( p T^{4} + \)\(13\!\cdots\!84\)\( p^{41} T^{5} + \)\(90\!\cdots\!12\)\( p^{82} T^{6} + \)\(39\!\cdots\!72\)\( p^{123} T^{7} + p^{164} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(36\!\cdots\!52\)\( T + \)\(23\!\cdots\!52\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} - \)\(59\!\cdots\!94\)\( T^{4} + \)\(31\!\cdots\!80\)\( p^{41} T^{5} + \)\(23\!\cdots\!52\)\( p^{82} T^{6} - \)\(36\!\cdots\!52\)\( p^{123} T^{7} + p^{164} T^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.90466941294547704442078424401, −10.08139757531473401389038677824, −9.544162973716166835633178740028, −9.487933126176598126347729864258, −9.340635324934120044071582964261, −8.631396541114538771867213723687, −8.168099141132219016509541559910, −8.119062495603604383653260599442, −7.57692701435250980805046229326, −7.21455050126391732412642137143, −6.64583078675594647141102048634, −6.09121522419422348833758769714, −5.67464253258325347012894904810, −4.71538140726844446009909166281, −4.51451479554662000466001581882, −4.16872280881164195584099576642, −4.16279728815865988996188367555, −3.27823885956949930655935892256, −2.80124302747572001597600018569, −2.49429621028374224008217175926, −1.97480057373067621755313571870, −1.91314548178943544225527390400, −1.24455453507659229966356001716, −0.967755440504959470005255867116, −0.23805643869082347528862747299, 0.23805643869082347528862747299, 0.967755440504959470005255867116, 1.24455453507659229966356001716, 1.91314548178943544225527390400, 1.97480057373067621755313571870, 2.49429621028374224008217175926, 2.80124302747572001597600018569, 3.27823885956949930655935892256, 4.16279728815865988996188367555, 4.16872280881164195584099576642, 4.51451479554662000466001581882, 4.71538140726844446009909166281, 5.67464253258325347012894904810, 6.09121522419422348833758769714, 6.64583078675594647141102048634, 7.21455050126391732412642137143, 7.57692701435250980805046229326, 8.119062495603604383653260599442, 8.168099141132219016509541559910, 8.631396541114538771867213723687, 9.340635324934120044071582964261, 9.487933126176598126347729864258, 9.544162973716166835633178740028, 10.08139757531473401389038677824, 10.90466941294547704442078424401

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