Properties

Label 8-2e16-1.1-c47e4-0-1
Degree $8$
Conductor $65536$
Sign $1$
Analytic cond. $2.51099\times 10^{9}$
Root an. cond. $14.9616$
Motivic weight $47$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.84e10·3-s − 3.11e16·5-s + 3.91e19·7-s − 6.09e22·9-s + 1.90e24·11-s + 1.26e26·13-s + 1.19e27·15-s + 2.10e29·17-s + 1.05e30·19-s − 1.50e30·21-s − 1.37e32·23-s − 4.32e32·25-s + 5.15e33·27-s − 2.27e34·29-s − 7.57e34·31-s − 7.31e34·33-s − 1.21e36·35-s − 1.12e37·37-s − 4.88e36·39-s + 1.30e38·41-s + 4.45e38·43-s + 1.89e39·45-s − 2.03e39·47-s − 5.14e39·49-s − 8.09e39·51-s + 2.91e40·53-s − 5.91e40·55-s + ⋯
L(s)  = 1  − 0.235·3-s − 1.16·5-s + 0.540·7-s − 2.29·9-s + 0.640·11-s + 0.843·13-s + 0.275·15-s + 2.55·17-s + 0.942·19-s − 0.127·21-s − 1.37·23-s − 0.608·25-s + 1.18·27-s − 0.978·29-s − 0.680·31-s − 0.151·33-s − 0.631·35-s − 1.58·37-s − 0.198·39-s + 1.64·41-s + 1.82·43-s + 2.67·45-s − 1.03·47-s − 0.980·49-s − 0.602·51-s + 0.879·53-s − 0.747·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+47/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Analytic conductor: \(2.51099\times 10^{9}\)
Root analytic conductor: \(14.9616\)
Motivic weight: \(47\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 65536,\ (\ :47/2, 47/2, 47/2, 47/2),\ 1)\)

Particular Values

\(L(24)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{49}{2})\) 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 \( 1 \)
good3$C_2 \wr S_4$ \( 1 + 4273499440 p^{2} T + 28556588122484284580 p^{7} T^{2} - \)\(94\!\cdots\!80\)\( p^{16} T^{3} + \)\(26\!\cdots\!74\)\( p^{27} T^{4} - \)\(94\!\cdots\!80\)\( p^{63} T^{5} + 28556588122484284580 p^{101} T^{6} + 4273499440 p^{143} T^{7} + p^{188} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 1244587209690888 p^{2} T + \)\(44\!\cdots\!08\)\( p^{5} T^{2} + \)\(32\!\cdots\!16\)\( p^{9} T^{3} + \)\(33\!\cdots\!66\)\( p^{16} T^{4} + \)\(32\!\cdots\!16\)\( p^{56} T^{5} + \)\(44\!\cdots\!08\)\( p^{99} T^{6} + 1244587209690888 p^{143} T^{7} + p^{188} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 799371810732416800 p^{2} T + \)\(27\!\cdots\!00\)\( p^{4} T^{2} - \)\(23\!\cdots\!00\)\( p^{6} T^{3} + \)\(11\!\cdots\!86\)\( p^{11} T^{4} - \)\(23\!\cdots\!00\)\( p^{53} T^{5} + \)\(27\!\cdots\!00\)\( p^{98} T^{6} - 799371810732416800 p^{143} T^{7} + p^{188} T^{8} \)
11$C_2 \wr S_4$ \( 1 - \)\(17\!\cdots\!92\)\( p T + \)\(20\!\cdots\!48\)\( p^{3} T^{2} - \)\(21\!\cdots\!24\)\( p^{6} T^{3} + \)\(14\!\cdots\!70\)\( p^{9} T^{4} - \)\(21\!\cdots\!24\)\( p^{53} T^{5} + \)\(20\!\cdots\!48\)\( p^{97} T^{6} - \)\(17\!\cdots\!92\)\( p^{142} T^{7} + p^{188} T^{8} \)
13$C_2 \wr S_4$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(51\!\cdots\!40\)\( p T^{2} - \)\(37\!\cdots\!20\)\( p^{3} T^{3} + \)\(41\!\cdots\!42\)\( p^{6} T^{4} - \)\(37\!\cdots\!20\)\( p^{50} T^{5} + \)\(51\!\cdots\!40\)\( p^{95} T^{6} - \)\(12\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \)
17$C_2 \wr S_4$ \( 1 - \)\(21\!\cdots\!80\)\( T + \)\(21\!\cdots\!20\)\( p T^{2} - \)\(82\!\cdots\!80\)\( p^{3} T^{3} + \)\(28\!\cdots\!94\)\( p^{5} T^{4} - \)\(82\!\cdots\!80\)\( p^{50} T^{5} + \)\(21\!\cdots\!20\)\( p^{95} T^{6} - \)\(21\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!40\)\( T + \)\(18\!\cdots\!24\)\( p T^{2} - \)\(46\!\cdots\!20\)\( p^{3} T^{3} + \)\(23\!\cdots\!74\)\( p^{5} T^{4} - \)\(46\!\cdots\!20\)\( p^{50} T^{5} + \)\(18\!\cdots\!24\)\( p^{95} T^{6} - \)\(10\!\cdots\!40\)\( p^{141} T^{7} + p^{188} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(59\!\cdots\!60\)\( p T + \)\(87\!\cdots\!20\)\( p^{2} T^{2} + \)\(34\!\cdots\!80\)\( p^{3} T^{3} + \)\(26\!\cdots\!98\)\( p^{4} T^{4} + \)\(34\!\cdots\!80\)\( p^{50} T^{5} + \)\(87\!\cdots\!20\)\( p^{96} T^{6} + \)\(59\!\cdots\!60\)\( p^{142} T^{7} + p^{188} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(22\!\cdots\!60\)\( T + \)\(46\!\cdots\!84\)\( p T^{2} + \)\(24\!\cdots\!20\)\( p^{2} T^{3} + \)\(33\!\cdots\!74\)\( p^{3} T^{4} + \)\(24\!\cdots\!20\)\( p^{49} T^{5} + \)\(46\!\cdots\!84\)\( p^{95} T^{6} + \)\(22\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \)
31$C_2 \wr S_4$ \( 1 + \)\(75\!\cdots\!48\)\( T + \)\(41\!\cdots\!68\)\( p T^{2} + \)\(42\!\cdots\!36\)\( p^{2} T^{3} + \)\(50\!\cdots\!70\)\( p^{3} T^{4} + \)\(42\!\cdots\!36\)\( p^{49} T^{5} + \)\(41\!\cdots\!68\)\( p^{95} T^{6} + \)\(75\!\cdots\!48\)\( p^{141} T^{7} + p^{188} T^{8} \)
37$C_2 \wr S_4$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(41\!\cdots\!60\)\( p T^{2} + \)\(68\!\cdots\!20\)\( p^{2} T^{3} + \)\(17\!\cdots\!26\)\( p^{3} T^{4} + \)\(68\!\cdots\!20\)\( p^{49} T^{5} + \)\(41\!\cdots\!60\)\( p^{95} T^{6} + \)\(11\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(31\!\cdots\!08\)\( p T + \)\(13\!\cdots\!28\)\( p^{2} T^{2} - \)\(30\!\cdots\!56\)\( p^{3} T^{3} + \)\(77\!\cdots\!70\)\( p^{4} T^{4} - \)\(30\!\cdots\!56\)\( p^{50} T^{5} + \)\(13\!\cdots\!28\)\( p^{96} T^{6} - \)\(31\!\cdots\!08\)\( p^{142} T^{7} + p^{188} T^{8} \)
43$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!00\)\( p T + \)\(13\!\cdots\!00\)\( p^{2} T^{2} - \)\(84\!\cdots\!00\)\( p^{3} T^{3} + \)\(61\!\cdots\!98\)\( p^{4} T^{4} - \)\(84\!\cdots\!00\)\( p^{50} T^{5} + \)\(13\!\cdots\!00\)\( p^{96} T^{6} - \)\(10\!\cdots\!00\)\( p^{142} T^{7} + p^{188} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(20\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(18\!\cdots\!60\)\( T^{3} + \)\(59\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!60\)\( p^{47} T^{5} + \)\(10\!\cdots\!60\)\( p^{94} T^{6} + \)\(20\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(29\!\cdots\!60\)\( T + \)\(33\!\cdots\!60\)\( T^{2} - \)\(87\!\cdots\!20\)\( T^{3} + \)\(50\!\cdots\!38\)\( T^{4} - \)\(87\!\cdots\!20\)\( p^{47} T^{5} + \)\(33\!\cdots\!60\)\( p^{94} T^{6} - \)\(29\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \)
59$C_2 \wr S_4$ \( 1 + \)\(47\!\cdots\!80\)\( T + \)\(71\!\cdots\!76\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!60\)\( p^{47} T^{5} + \)\(71\!\cdots\!76\)\( p^{94} T^{6} + \)\(47\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(62\!\cdots\!88\)\( T + \)\(29\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!36\)\( p^{47} T^{5} + \)\(29\!\cdots\!88\)\( p^{94} T^{6} - \)\(62\!\cdots\!88\)\( p^{141} T^{7} + p^{188} T^{8} \)
67$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!80\)\( T + \)\(35\!\cdots\!40\)\( T^{2} + \)\(38\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} + \)\(38\!\cdots\!40\)\( p^{47} T^{5} + \)\(35\!\cdots\!40\)\( p^{94} T^{6} + \)\(18\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \)
71$C_2 \wr S_4$ \( 1 + \)\(22\!\cdots\!68\)\( T + \)\(32\!\cdots\!48\)\( T^{2} + \)\(42\!\cdots\!16\)\( T^{3} + \)\(43\!\cdots\!70\)\( T^{4} + \)\(42\!\cdots\!16\)\( p^{47} T^{5} + \)\(32\!\cdots\!48\)\( p^{94} T^{6} + \)\(22\!\cdots\!68\)\( p^{141} T^{7} + p^{188} T^{8} \)
73$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(14\!\cdots\!80\)\( T^{2} - \)\(90\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!18\)\( T^{4} - \)\(90\!\cdots\!60\)\( p^{47} T^{5} + \)\(14\!\cdots\!80\)\( p^{94} T^{6} - \)\(10\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(13\!\cdots\!60\)\( T + \)\(11\!\cdots\!36\)\( T^{2} - \)\(68\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} - \)\(68\!\cdots\!20\)\( p^{47} T^{5} + \)\(11\!\cdots\!36\)\( p^{94} T^{6} - \)\(13\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!60\)\( T + \)\(39\!\cdots\!40\)\( T^{2} - \)\(29\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!58\)\( T^{4} - \)\(29\!\cdots\!20\)\( p^{47} T^{5} + \)\(39\!\cdots\!40\)\( p^{94} T^{6} - \)\(14\!\cdots\!60\)\( p^{141} T^{7} + p^{188} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(79\!\cdots\!80\)\( T + \)\(72\!\cdots\!16\)\( T^{2} + \)\(58\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!46\)\( T^{4} + \)\(58\!\cdots\!60\)\( p^{47} T^{5} + \)\(72\!\cdots\!16\)\( p^{94} T^{6} + \)\(79\!\cdots\!80\)\( p^{141} T^{7} + p^{188} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(95\!\cdots\!20\)\( T + \)\(11\!\cdots\!60\)\( T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!38\)\( T^{4} - \)\(65\!\cdots\!60\)\( p^{47} T^{5} + \)\(11\!\cdots\!60\)\( p^{94} T^{6} - \)\(95\!\cdots\!20\)\( p^{141} T^{7} + p^{188} T^{8} \)
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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

−7.947698563235009285742006197398, −7.35548906665726781824642605964, −7.34239099696843631330450010928, −6.63593876006304542149653767212, −6.52655305753429207436091411310, −5.96791777002274721451807616827, −5.91088649146271843248510197561, −5.57487579639022694153028811429, −5.50289084213880659421386611083, −5.25828221232195281446378604367, −4.66633033113696054641368924680, −4.60717069128014041927744758669, −4.08120357435325617258550795575, −3.67057336071345396925854769165, −3.54605361153948818087271262260, −3.44379347655968321394455145134, −3.34630890604671075809091529738, −2.81000190041243396312116947255, −2.33910256312820722101447236637, −2.26938615685663392446568416324, −2.02140298717201603525385995025, −1.28273989758719814630566446861, −1.15733683538810549237334056439, −1.11158832166714735986795392214, −0.911690454730985764622225795330, 0, 0, 0, 0, 0.911690454730985764622225795330, 1.11158832166714735986795392214, 1.15733683538810549237334056439, 1.28273989758719814630566446861, 2.02140298717201603525385995025, 2.26938615685663392446568416324, 2.33910256312820722101447236637, 2.81000190041243396312116947255, 3.34630890604671075809091529738, 3.44379347655968321394455145134, 3.54605361153948818087271262260, 3.67057336071345396925854769165, 4.08120357435325617258550795575, 4.60717069128014041927744758669, 4.66633033113696054641368924680, 5.25828221232195281446378604367, 5.50289084213880659421386611083, 5.57487579639022694153028811429, 5.91088649146271843248510197561, 5.96791777002274721451807616827, 6.52655305753429207436091411310, 6.63593876006304542149653767212, 7.34239099696843631330450010928, 7.35548906665726781824642605964, 7.947698563235009285742006197398

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