Properties

Label 8-3e8-1.1-c53e4-0-1
Degree $8$
Conductor $6561$
Sign $1$
Analytic cond. $6.57210\times 10^{8}$
Root an. cond. $12.6535$
Motivic weight $53$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8.04e7·2-s − 5.36e15·4-s − 4.30e18·5-s − 1.58e22·7-s + 7.67e23·8-s + 3.46e26·10-s + 1.69e27·11-s − 5.75e29·13-s + 1.27e30·14-s − 5.28e31·16-s − 3.63e32·17-s + 1.86e34·19-s + 2.31e34·20-s − 1.36e35·22-s + 1.65e36·23-s − 1.98e37·25-s + 4.62e37·26-s + 8.50e37·28-s + 3.02e38·29-s + 2.37e39·31-s + 1.30e39·32-s + 2.92e40·34-s + 6.82e40·35-s + 3.26e41·37-s − 1.50e42·38-s − 3.30e42·40-s + 1.14e43·41-s + ⋯
L(s)  = 1  − 0.847·2-s − 0.596·4-s − 1.29·5-s − 0.637·7-s + 0.898·8-s + 1.09·10-s + 0.429·11-s − 1.73·13-s + 0.540·14-s − 0.651·16-s − 0.898·17-s + 2.42·19-s + 0.770·20-s − 0.363·22-s + 1.35·23-s − 1.78·25-s + 1.47·26-s + 0.380·28-s + 0.533·29-s + 0.714·31-s + 0.169·32-s + 0.761·34-s + 0.824·35-s + 0.904·37-s − 2.05·38-s − 1.16·40-s + 2.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(54-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+53/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6561\)    =    \(3^{8}\)
Sign: $1$
Analytic conductor: \(6.57210\times 10^{8}\)
Root analytic conductor: \(12.6535\)
Motivic weight: \(53\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6561,\ (\ :53/2, 53/2, 53/2, 53/2),\ 1)\)

Particular Values

\(L(27)\) \(\approx\) \(3.075468377\)
\(L(\frac12)\) \(\approx\) \(3.075468377\)
\(L(\frac{55}{2})\) 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)$
bad3 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 20107737 p^{2} T + 46246123451435 p^{8} T^{2} + 2351569503320792703 p^{18} T^{3} + \)\(12\!\cdots\!03\)\( p^{33} T^{4} + 2351569503320792703 p^{71} T^{5} + 46246123451435 p^{114} T^{6} + 20107737 p^{161} T^{7} + p^{212} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 861320362273981848 p T + \)\(61\!\cdots\!08\)\( p^{4} T^{2} + \)\(31\!\cdots\!04\)\( p^{8} T^{3} + \)\(10\!\cdots\!26\)\( p^{14} T^{4} + \)\(31\!\cdots\!04\)\( p^{61} T^{5} + \)\(61\!\cdots\!08\)\( p^{110} T^{6} + 861320362273981848 p^{160} T^{7} + p^{212} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 46180366430630158528 p^{3} T + \)\(14\!\cdots\!88\)\( p^{6} T^{2} + \)\(76\!\cdots\!08\)\( p^{11} T^{3} + \)\(38\!\cdots\!70\)\( p^{16} T^{4} + \)\(76\!\cdots\!08\)\( p^{64} T^{5} + \)\(14\!\cdots\!88\)\( p^{112} T^{6} + 46180366430630158528 p^{162} T^{7} + p^{212} T^{8} \)
11$C_2 \wr S_4$ \( 1 - \)\(15\!\cdots\!64\)\( p T + \)\(13\!\cdots\!72\)\( p^{2} T^{2} + \)\(32\!\cdots\!60\)\( p^{3} T^{3} - \)\(26\!\cdots\!74\)\( p^{6} T^{4} + \)\(32\!\cdots\!60\)\( p^{56} T^{5} + \)\(13\!\cdots\!72\)\( p^{108} T^{6} - \)\(15\!\cdots\!64\)\( p^{160} T^{7} + p^{212} T^{8} \)
13$C_2 \wr S_4$ \( 1 + \)\(44\!\cdots\!56\)\( p T + \)\(16\!\cdots\!12\)\( p^{3} T^{2} + \)\(33\!\cdots\!52\)\( p^{5} T^{3} + \)\(85\!\cdots\!50\)\( p^{7} T^{4} + \)\(33\!\cdots\!52\)\( p^{58} T^{5} + \)\(16\!\cdots\!12\)\( p^{109} T^{6} + \)\(44\!\cdots\!56\)\( p^{160} T^{7} + p^{212} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(36\!\cdots\!32\)\( T + \)\(28\!\cdots\!96\)\( p T^{2} + \)\(19\!\cdots\!00\)\( p^{3} T^{3} + \)\(68\!\cdots\!18\)\( p^{5} T^{4} + \)\(19\!\cdots\!00\)\( p^{56} T^{5} + \)\(28\!\cdots\!96\)\( p^{107} T^{6} + \)\(36\!\cdots\!32\)\( p^{159} T^{7} + p^{212} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(18\!\cdots\!88\)\( T + \)\(17\!\cdots\!28\)\( p T^{2} - \)\(49\!\cdots\!24\)\( p^{3} T^{3} + \)\(12\!\cdots\!86\)\( p^{5} T^{4} - \)\(49\!\cdots\!24\)\( p^{56} T^{5} + \)\(17\!\cdots\!28\)\( p^{107} T^{6} - \)\(18\!\cdots\!88\)\( p^{159} T^{7} + p^{212} T^{8} \)
23$C_2 \wr S_4$ \( 1 - \)\(16\!\cdots\!92\)\( T + \)\(99\!\cdots\!56\)\( p T^{2} - \)\(70\!\cdots\!12\)\( p^{2} T^{3} + \)\(24\!\cdots\!30\)\( p^{4} T^{4} - \)\(70\!\cdots\!12\)\( p^{55} T^{5} + \)\(99\!\cdots\!56\)\( p^{107} T^{6} - \)\(16\!\cdots\!92\)\( p^{159} T^{7} + p^{212} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!64\)\( T + \)\(48\!\cdots\!00\)\( T^{2} - \)\(20\!\cdots\!28\)\( T^{3} + \)\(86\!\cdots\!02\)\( p T^{4} - \)\(20\!\cdots\!28\)\( p^{53} T^{5} + \)\(48\!\cdots\!00\)\( p^{106} T^{6} - \)\(30\!\cdots\!64\)\( p^{159} T^{7} + p^{212} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(23\!\cdots\!24\)\( T + \)\(52\!\cdots\!68\)\( p T^{2} - \)\(58\!\cdots\!32\)\( p^{2} T^{3} + \)\(59\!\cdots\!34\)\( p^{3} T^{4} - \)\(58\!\cdots\!32\)\( p^{55} T^{5} + \)\(52\!\cdots\!68\)\( p^{107} T^{6} - \)\(23\!\cdots\!24\)\( p^{159} T^{7} + p^{212} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(32\!\cdots\!96\)\( T + \)\(87\!\cdots\!96\)\( p T^{2} - \)\(45\!\cdots\!64\)\( p^{2} T^{3} + \)\(94\!\cdots\!90\)\( p^{3} T^{4} - \)\(45\!\cdots\!64\)\( p^{55} T^{5} + \)\(87\!\cdots\!96\)\( p^{107} T^{6} - \)\(32\!\cdots\!96\)\( p^{159} T^{7} + p^{212} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(11\!\cdots\!56\)\( T + \)\(12\!\cdots\!68\)\( T^{2} - \)\(20\!\cdots\!08\)\( p T^{3} + \)\(33\!\cdots\!54\)\( p^{2} T^{4} - \)\(20\!\cdots\!08\)\( p^{54} T^{5} + \)\(12\!\cdots\!68\)\( p^{106} T^{6} - \)\(11\!\cdots\!56\)\( p^{159} T^{7} + p^{212} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(43\!\cdots\!00\)\( p T + \)\(69\!\cdots\!40\)\( p^{2} T^{2} + \)\(18\!\cdots\!00\)\( p^{3} T^{3} + \)\(19\!\cdots\!98\)\( p^{4} T^{4} + \)\(18\!\cdots\!00\)\( p^{56} T^{5} + \)\(69\!\cdots\!40\)\( p^{108} T^{6} + \)\(43\!\cdots\!00\)\( p^{160} T^{7} + p^{212} T^{8} \)
47$C_2 \wr S_4$ \( 1 - \)\(35\!\cdots\!20\)\( T + \)\(32\!\cdots\!60\)\( p T^{2} - \)\(15\!\cdots\!60\)\( p^{2} T^{3} + \)\(88\!\cdots\!46\)\( p^{3} T^{4} - \)\(15\!\cdots\!60\)\( p^{55} T^{5} + \)\(32\!\cdots\!60\)\( p^{107} T^{6} - \)\(35\!\cdots\!20\)\( p^{159} T^{7} + p^{212} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!52\)\( T + \)\(41\!\cdots\!88\)\( T^{2} - \)\(16\!\cdots\!08\)\( T^{3} + \)\(92\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!08\)\( p^{53} T^{5} + \)\(41\!\cdots\!88\)\( p^{106} T^{6} - \)\(30\!\cdots\!52\)\( p^{159} T^{7} + p^{212} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(12\!\cdots\!68\)\( T + \)\(12\!\cdots\!72\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + \)\(72\!\cdots\!94\)\( T^{4} - \)\(11\!\cdots\!16\)\( p^{53} T^{5} + \)\(12\!\cdots\!72\)\( p^{106} T^{6} - \)\(12\!\cdots\!68\)\( p^{159} T^{7} + p^{212} T^{8} \)
61$C_2 \wr S_4$ \( 1 + \)\(48\!\cdots\!60\)\( T + \)\(14\!\cdots\!76\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(41\!\cdots\!40\)\( p^{53} T^{5} + \)\(14\!\cdots\!76\)\( p^{106} T^{6} + \)\(48\!\cdots\!60\)\( p^{159} T^{7} + p^{212} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!32\)\( T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!26\)\( T^{4} - \)\(21\!\cdots\!00\)\( p^{53} T^{5} + \)\(10\!\cdots\!32\)\( p^{106} T^{6} - \)\(14\!\cdots\!32\)\( p^{159} T^{7} + p^{212} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(91\!\cdots\!72\)\( T + \)\(30\!\cdots\!88\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(46\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!04\)\( p^{53} T^{5} + \)\(30\!\cdots\!88\)\( p^{106} T^{6} - \)\(91\!\cdots\!72\)\( p^{159} T^{7} + p^{212} T^{8} \)
73$C_2 \wr S_4$ \( 1 - \)\(56\!\cdots\!28\)\( T + \)\(17\!\cdots\!68\)\( T^{2} - \)\(20\!\cdots\!32\)\( T^{3} + \)\(27\!\cdots\!10\)\( T^{4} - \)\(20\!\cdots\!32\)\( p^{53} T^{5} + \)\(17\!\cdots\!68\)\( p^{106} T^{6} - \)\(56\!\cdots\!28\)\( p^{159} T^{7} + p^{212} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(34\!\cdots\!40\)\( p T + \)\(14\!\cdots\!56\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!26\)\( T^{4} - \)\(28\!\cdots\!20\)\( p^{53} T^{5} + \)\(14\!\cdots\!56\)\( p^{106} T^{6} - \)\(34\!\cdots\!40\)\( p^{160} T^{7} + p^{212} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(14\!\cdots\!44\)\( T + \)\(23\!\cdots\!20\)\( T^{2} + \)\(19\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!64\)\( p^{53} T^{5} + \)\(23\!\cdots\!20\)\( p^{106} T^{6} + \)\(14\!\cdots\!44\)\( p^{159} T^{7} + p^{212} T^{8} \)
89$C_2 \wr S_4$ \( 1 - \)\(11\!\cdots\!32\)\( T + \)\(98\!\cdots\!92\)\( T^{2} - \)\(57\!\cdots\!84\)\( T^{3} + \)\(28\!\cdots\!74\)\( T^{4} - \)\(57\!\cdots\!84\)\( p^{53} T^{5} + \)\(98\!\cdots\!92\)\( p^{106} T^{6} - \)\(11\!\cdots\!32\)\( p^{159} T^{7} + p^{212} T^{8} \)
97$C_2 \wr S_4$ \( 1 + \)\(61\!\cdots\!48\)\( T + \)\(53\!\cdots\!72\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!66\)\( T^{4} + \)\(26\!\cdots\!00\)\( p^{53} T^{5} + \)\(53\!\cdots\!72\)\( p^{106} T^{6} + \)\(61\!\cdots\!48\)\( p^{159} T^{7} + p^{212} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.55163725247562718724625685169, −7.34774198009336572527476581888, −7.16751324420932738265341109256, −6.66095338456805619861862937739, −6.47782096139603487232004297905, −6.12905015290237894028561914746, −5.59460075403141818428256978565, −5.27108644918727664165148475765, −5.11529439443746148483610322890, −4.71801327156031941465799536213, −4.37888914669463092047828292931, −4.19986397064041699594119250582, −3.99386322851232775057558364379, −3.40258663334745637042514384562, −3.38815653164938543962886502012, −2.88446114350568324765868887400, −2.63968438433604599188021391600, −2.43287537229291703473135819865, −1.84691926826776824235590661824, −1.71054274199201372712995926239, −1.28217954101213675192678441643, −0.59875171863732607502533095333, −0.57738946496209304162664262030, −0.48956585243696326250064183685, −0.48458625173991755401789359508, 0.48458625173991755401789359508, 0.48956585243696326250064183685, 0.57738946496209304162664262030, 0.59875171863732607502533095333, 1.28217954101213675192678441643, 1.71054274199201372712995926239, 1.84691926826776824235590661824, 2.43287537229291703473135819865, 2.63968438433604599188021391600, 2.88446114350568324765868887400, 3.38815653164938543962886502012, 3.40258663334745637042514384562, 3.99386322851232775057558364379, 4.19986397064041699594119250582, 4.37888914669463092047828292931, 4.71801327156031941465799536213, 5.11529439443746148483610322890, 5.27108644918727664165148475765, 5.59460075403141818428256978565, 6.12905015290237894028561914746, 6.47782096139603487232004297905, 6.66095338456805619861862937739, 7.16751324420932738265341109256, 7.34774198009336572527476581888, 7.55163725247562718724625685169

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