Properties

Degree 8
Conductor $ 2^{12} $
Sign $1$
Motivic weight 6
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 48·3-s − 20·4-s − 96·6-s + 40·8-s − 636·9-s + 976·11-s + 960·12-s − 3.28e3·16-s + 4.16e3·17-s − 1.27e3·18-s − 1.45e3·19-s + 1.95e3·22-s − 1.92e3·24-s + 1.93e4·25-s + 7.99e4·27-s − 1.82e4·32-s − 4.68e4·33-s + 8.33e3·34-s + 1.27e4·36-s − 2.91e3·38-s − 1.17e5·41-s + 1.97e5·43-s − 1.95e4·44-s + 1.57e5·48-s + 2.36e5·49-s + 3.86e4·50-s + ⋯
L(s)  = 1  + 1/4·2-s − 1.77·3-s − 0.312·4-s − 4/9·6-s + 5/64·8-s − 0.872·9-s + 0.733·11-s + 5/9·12-s − 0.800·16-s + 0.848·17-s − 0.218·18-s − 0.212·19-s + 0.183·22-s − 0.138·24-s + 1.23·25-s + 4.06·27-s − 0.557·32-s − 1.30·33-s + 0.212·34-s + 0.272·36-s − 0.0530·38-s − 1.71·41-s + 2.48·43-s − 0.229·44-s + 1.42·48-s + 2.00·49-s + 0.308·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(4096\)    =    \(2^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  induced by $\chi_{8} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 4096,\ (\ :3, 3, 3, 3),\ 1)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.589912\)
\(L(\frac12)\)  \(\approx\)  \(0.589912\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 8. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$D_{4}$ \( 1 - p T + 3 p^{3} T^{2} - p^{7} T^{3} + p^{12} T^{4} \)
good3$D_{4}$ \( ( 1 + 8 p T + 394 p T^{2} + 8 p^{7} T^{3} + p^{12} T^{4} )^{2} \)
5$C_2^2 \wr C_2$ \( 1 - 772 p^{2} T^{2} + 2049246 p^{3} T^{4} - 772 p^{14} T^{6} + p^{24} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 236356 T^{2} + 28021959366 T^{4} - 236356 p^{12} T^{6} + p^{24} T^{8} \)
11$D_{4}$ \( ( 1 - 488 T + 2238078 T^{2} - 488 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 - 4994596 T^{2} + 30260873415846 T^{4} - 4994596 p^{12} T^{6} + p^{24} T^{8} \)
17$D_{4}$ \( ( 1 - 2084 T + 32560902 T^{2} - 2084 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 728 T + 92449758 T^{2} + 728 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 212117956 T^{2} + 1001159632705962 p T^{4} - 212117956 p^{12} T^{6} + p^{24} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - 1409719204 p^{12} T^{6} + p^{24} T^{8} \)
31$C_2^2 \wr C_2$ \( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - 2758360324 p^{12} T^{6} + p^{24} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - 8121202276 p^{12} T^{6} + p^{24} T^{8} \)
41$D_{4}$ \( ( 1 + 58972 T + 240432438 p T^{2} + 58972 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 2296 p T + 13190101374 T^{2} - 2296 p^{7} T^{3} + p^{12} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13578767236 p^{12} T^{6} + p^{24} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - 42194545636 p^{12} T^{6} + p^{24} T^{8} \)
59$D_{4}$ \( ( 1 - 271016 T + 102678575166 T^{2} - 271016 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 - 2637195124 p T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - 2637195124 p^{13} T^{6} + p^{24} T^{8} \)
67$D_{4}$ \( ( 1 + 395096 T + 204987839262 T^{2} + 395096 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - 164364621124 p^{12} T^{6} + p^{24} T^{8} \)
73$D_{4}$ \( ( 1 - 221956 T + 284381984742 T^{2} - 221956 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - 828257339524 p^{12} T^{6} + p^{24} T^{8} \)
83$D_{4}$ \( ( 1 + 1732504 T + 1400265667422 T^{2} + 1732504 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 380612 T + 970422538278 T^{2} - 380612 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 463388 T + 953987784774 T^{2} + 463388 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
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

−15.57600905421142132280584557032, −15.33488768624734424137414583545, −14.44910702253910699477665917260, −14.39316917820394788122429598033, −14.12461614241560506729601217816, −13.60154628238059635476301142235, −12.90432594182836474830169553867, −12.55555373117196914310731018643, −12.04150628738107464748147852586, −11.64637614916148065835557843715, −11.33866024450355041823754287163, −11.14144158935495434278459082971, −10.35341239530777941003521327660, −10.17149663938200605468979743911, −9.055539380151302611188845939990, −8.813205526723579910742409612834, −8.412656806276341346209563536903, −7.28196052882993323425064481501, −6.81522779244751174684062919760, −5.83390581301779880453249431197, −5.80367130656092429931878280249, −5.08955307076141463251605139114, −4.20051198457975790597031324160, −2.86446596448982736560703191197, −0.59900738430399026808425440808, 0.59900738430399026808425440808, 2.86446596448982736560703191197, 4.20051198457975790597031324160, 5.08955307076141463251605139114, 5.80367130656092429931878280249, 5.83390581301779880453249431197, 6.81522779244751174684062919760, 7.28196052882993323425064481501, 8.412656806276341346209563536903, 8.813205526723579910742409612834, 9.055539380151302611188845939990, 10.17149663938200605468979743911, 10.35341239530777941003521327660, 11.14144158935495434278459082971, 11.33866024450355041823754287163, 11.64637614916148065835557843715, 12.04150628738107464748147852586, 12.55555373117196914310731018643, 12.90432594182836474830169553867, 13.60154628238059635476301142235, 14.12461614241560506729601217816, 14.39316917820394788122429598033, 14.44910702253910699477665917260, 15.33488768624734424137414583545, 15.57600905421142132280584557032

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