Properties

Degree 8
Conductor $ 3^{8} $
Sign $1$
Motivic weight 37
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4.37e5·2-s − 6.09e9·4-s + 4.09e12·5-s + 6.60e15·7-s + 3.81e15·8-s − 1.79e18·10-s − 2.09e19·11-s + 5.18e19·13-s − 2.89e21·14-s + 1.05e22·16-s − 8.11e22·17-s − 5.44e23·19-s − 2.50e22·20-s + 9.16e24·22-s + 6.10e24·23-s − 8.94e25·25-s − 2.26e25·26-s − 4.02e25·28-s − 4.17e27·29-s + 8.99e27·31-s − 5.09e27·32-s + 3.54e28·34-s + 2.70e28·35-s + 5.55e28·37-s + 2.38e29·38-s + 1.56e28·40-s + 8.69e29·41-s + ⋯
L(s)  = 1  − 1.18·2-s − 0.0443·4-s + 0.480·5-s + 1.53·7-s + 0.0748·8-s − 0.567·10-s − 1.13·11-s + 0.127·13-s − 1.80·14-s + 0.560·16-s − 1.39·17-s − 1.19·19-s − 0.0213·20-s + 1.34·22-s + 0.392·23-s − 1.22·25-s − 0.150·26-s − 0.0680·28-s − 3.68·29-s + 2.31·31-s − 0.727·32-s + 1.65·34-s + 0.736·35-s + 0.540·37-s + 1.41·38-s + 0.0359·40-s + 1.26·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+37/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(6561\)    =    \(3^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  induced by $\chi_{9} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 6561,\ (\ :37/2, 37/2, 37/2, 37/2),\ 1)$
$L(19)$  $\approx$  $0.9061870267$
$L(\frac12)$  $\approx$  $0.9061870267$
$L(\frac{39}{2})$   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 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 218781 p T + 6173708785 p^{5} T^{2} + 2603177122851 p^{15} T^{3} + 783039221777613 p^{25} T^{4} + 2603177122851 p^{52} T^{5} + 6173708785 p^{79} T^{6} + 218781 p^{112} T^{7} + p^{148} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 4099829756904 T + \)\(84\!\cdots\!16\)\( p^{3} T^{2} + \)\(16\!\cdots\!48\)\( p^{8} T^{3} + \)\(34\!\cdots\!82\)\( p^{14} T^{4} + \)\(16\!\cdots\!48\)\( p^{45} T^{5} + \)\(84\!\cdots\!16\)\( p^{77} T^{6} - 4099829756904 p^{111} T^{7} + p^{148} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 6605809948153184 T + \)\(55\!\cdots\!76\)\( p T^{2} - \)\(69\!\cdots\!44\)\( p^{4} T^{3} + \)\(65\!\cdots\!50\)\( p^{7} T^{4} - \)\(69\!\cdots\!44\)\( p^{41} T^{5} + \)\(55\!\cdots\!76\)\( p^{75} T^{6} - 6605809948153184 p^{111} T^{7} + p^{148} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 1904882622926844816 p T + \)\(76\!\cdots\!52\)\( p^{2} T^{2} + \)\(80\!\cdots\!20\)\( p^{5} T^{3} + \)\(19\!\cdots\!26\)\( p^{8} T^{4} + \)\(80\!\cdots\!20\)\( p^{42} T^{5} + \)\(76\!\cdots\!52\)\( p^{76} T^{6} + 1904882622926844816 p^{112} T^{7} + p^{148} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 51830892788989874168 T + \)\(47\!\cdots\!04\)\( T^{2} - \)\(13\!\cdots\!52\)\( p T^{3} + \)\(48\!\cdots\!50\)\( p^{3} T^{4} - \)\(13\!\cdots\!52\)\( p^{38} T^{5} + \)\(47\!\cdots\!04\)\( p^{74} T^{6} - 51830892788989874168 p^{111} T^{7} + p^{148} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(81\!\cdots\!28\)\( T + \)\(59\!\cdots\!56\)\( p T^{2} + \)\(14\!\cdots\!80\)\( p^{3} T^{3} + \)\(31\!\cdots\!18\)\( p^{5} T^{4} + \)\(14\!\cdots\!80\)\( p^{40} T^{5} + \)\(59\!\cdots\!56\)\( p^{75} T^{6} + \)\(81\!\cdots\!28\)\( p^{111} T^{7} + p^{148} T^{8} \)
19$C_2 \wr S_4$ \( 1 + \)\(54\!\cdots\!32\)\( T + \)\(55\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!16\)\( p T^{3} + \)\(44\!\cdots\!34\)\( p^{2} T^{4} + \)\(13\!\cdots\!16\)\( p^{38} T^{5} + \)\(55\!\cdots\!72\)\( p^{74} T^{6} + \)\(54\!\cdots\!32\)\( p^{111} T^{7} + p^{148} T^{8} \)
23$C_2 \wr S_4$ \( 1 - \)\(61\!\cdots\!88\)\( T + \)\(42\!\cdots\!16\)\( p T^{2} - \)\(32\!\cdots\!16\)\( p^{3} T^{3} + \)\(16\!\cdots\!70\)\( p^{3} T^{4} - \)\(32\!\cdots\!16\)\( p^{40} T^{5} + \)\(42\!\cdots\!16\)\( p^{75} T^{6} - \)\(61\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(41\!\cdots\!36\)\( T + \)\(35\!\cdots\!20\)\( p T^{2} + \)\(20\!\cdots\!52\)\( p^{2} T^{3} + \)\(89\!\cdots\!02\)\( p^{3} T^{4} + \)\(20\!\cdots\!52\)\( p^{39} T^{5} + \)\(35\!\cdots\!20\)\( p^{75} T^{6} + \)\(41\!\cdots\!36\)\( p^{111} T^{7} + p^{148} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(89\!\cdots\!64\)\( T + \)\(23\!\cdots\!28\)\( p T^{2} - \)\(38\!\cdots\!92\)\( p^{2} T^{3} + \)\(57\!\cdots\!94\)\( p^{3} T^{4} - \)\(38\!\cdots\!92\)\( p^{39} T^{5} + \)\(23\!\cdots\!28\)\( p^{75} T^{6} - \)\(89\!\cdots\!64\)\( p^{111} T^{7} + p^{148} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(55\!\cdots\!24\)\( T + \)\(22\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(20\!\cdots\!44\)\( p^{37} T^{5} + \)\(22\!\cdots\!72\)\( p^{74} T^{6} - \)\(55\!\cdots\!24\)\( p^{111} T^{7} + p^{148} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(86\!\cdots\!76\)\( T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(10\!\cdots\!28\)\( p^{37} T^{5} + \)\(17\!\cdots\!88\)\( p^{74} T^{6} - \)\(86\!\cdots\!76\)\( p^{111} T^{7} + p^{148} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(50\!\cdots\!80\)\( T + \)\(85\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(56\!\cdots\!40\)\( p^{37} T^{5} + \)\(85\!\cdots\!80\)\( p^{74} T^{6} + \)\(50\!\cdots\!80\)\( p^{111} T^{7} + p^{148} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(42\!\cdots\!20\)\( T + \)\(24\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!40\)\( p^{37} T^{5} + \)\(24\!\cdots\!20\)\( p^{74} T^{6} + \)\(42\!\cdots\!20\)\( p^{111} T^{7} + p^{148} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!88\)\( T^{2} - \)\(19\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} - \)\(19\!\cdots\!32\)\( p^{37} T^{5} + \)\(20\!\cdots\!88\)\( p^{74} T^{6} - \)\(12\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(13\!\cdots\!88\)\( T + \)\(62\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{3} - \)\(29\!\cdots\!66\)\( T^{4} + \)\(19\!\cdots\!84\)\( p^{37} T^{5} + \)\(62\!\cdots\!12\)\( p^{74} T^{6} - \)\(13\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
61$C_2 \wr S_4$ \( 1 + \)\(12\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!46\)\( T^{4} + \)\(36\!\cdots\!40\)\( p^{37} T^{5} + \)\(43\!\cdots\!16\)\( p^{74} T^{6} + \)\(12\!\cdots\!60\)\( p^{111} T^{7} + p^{148} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(16\!\cdots\!48\)\( T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!00\)\( p^{37} T^{5} + \)\(20\!\cdots\!72\)\( p^{74} T^{6} - \)\(16\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
71$C_2 \wr S_4$ \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(55\!\cdots\!68\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} + \)\(47\!\cdots\!16\)\( p^{37} T^{5} + \)\(55\!\cdots\!68\)\( p^{74} T^{6} + \)\(10\!\cdots\!88\)\( p^{111} T^{7} + p^{148} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(19\!\cdots\!48\)\( T + \)\(28\!\cdots\!68\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + \)\(45\!\cdots\!52\)\( p^{37} T^{5} + \)\(28\!\cdots\!68\)\( p^{74} T^{6} + \)\(19\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(42\!\cdots\!20\)\( T + \)\(83\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} - \)\(27\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{37} T^{5} + \)\(83\!\cdots\!36\)\( p^{74} T^{6} - \)\(42\!\cdots\!20\)\( p^{111} T^{7} + p^{148} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(46\!\cdots\!24\)\( T + \)\(47\!\cdots\!60\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!44\)\( p^{37} T^{5} + \)\(47\!\cdots\!60\)\( p^{74} T^{6} - \)\(46\!\cdots\!24\)\( p^{111} T^{7} + p^{148} T^{8} \)
89$C_2 \wr S_4$ \( 1 - \)\(31\!\cdots\!52\)\( T + \)\(35\!\cdots\!52\)\( T^{2} - \)\(89\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} - \)\(89\!\cdots\!84\)\( p^{37} T^{5} + \)\(35\!\cdots\!52\)\( p^{74} T^{6} - \)\(31\!\cdots\!52\)\( p^{111} T^{7} + p^{148} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(44\!\cdots\!48\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!40\)\( p^{37} T^{5} + \)\(13\!\cdots\!12\)\( p^{74} T^{6} - \)\(44\!\cdots\!48\)\( p^{111} T^{7} + p^{148} T^{8} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.757942440246284617315481444702, −8.198214539228176137496746117651, −8.182242227282327811052116258004, −7.87511309095839839874909499433, −7.54720713749742113369378399355, −6.95643697468195119764446931995, −6.81802612383089184368840713825, −6.10880970572664590835727833554, −5.93993614889857618017393582226, −5.59795792675660459153752481295, −5.16107299421062086737074944040, −4.87716626848294671609516296283, −4.61326015245259160371731162947, −3.99770135500324232484230150378, −3.87800239795090749687210611690, −3.57770801743257322787710755178, −2.75345715904718220366226058916, −2.36713612551349998305232084055, −2.36567096958845444449601641358, −1.80289714052451018836049793860, −1.78182685151025830588465770767, −1.20008575568362234212282145998, −0.73282083784995701345709659954, −0.44934044528701019307714488010, −0.21108633911225696834721495038, 0.21108633911225696834721495038, 0.44934044528701019307714488010, 0.73282083784995701345709659954, 1.20008575568362234212282145998, 1.78182685151025830588465770767, 1.80289714052451018836049793860, 2.36567096958845444449601641358, 2.36713612551349998305232084055, 2.75345715904718220366226058916, 3.57770801743257322787710755178, 3.87800239795090749687210611690, 3.99770135500324232484230150378, 4.61326015245259160371731162947, 4.87716626848294671609516296283, 5.16107299421062086737074944040, 5.59795792675660459153752481295, 5.93993614889857618017393582226, 6.10880970572664590835727833554, 6.81802612383089184368840713825, 6.95643697468195119764446931995, 7.54720713749742113369378399355, 7.87511309095839839874909499433, 8.182242227282327811052116258004, 8.198214539228176137496746117651, 8.757942440246284617315481444702

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