Properties

Label 8-1-1.1-c53e4-0-0
Degree $8$
Conductor $1$
Sign $1$
Analytic cond. $100169.$
Root an. cond. $4.21785$
Motivic weight $53$
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  − 6.84e7·2-s − 1.04e12·3-s − 1.17e16·4-s − 4.56e18·5-s + 7.17e19·6-s − 2.24e20·7-s + 1.16e24·8-s − 3.25e25·9-s + 3.12e26·10-s + 2.48e27·11-s + 1.23e28·12-s + 3.88e28·13-s + 1.53e28·14-s + 4.78e30·15-s − 7.29e30·16-s − 8.69e32·17-s + 2.22e33·18-s − 2.55e34·19-s + 5.36e34·20-s + 2.35e32·21-s − 1.69e35·22-s + 6.92e35·23-s − 1.22e36·24-s + 6.45e36·25-s − 2.66e36·26-s + 6.96e37·27-s + 2.64e36·28-s + ⋯
L(s)  = 1  − 0.721·2-s − 0.238·3-s − 1.30·4-s − 1.36·5-s + 0.171·6-s − 0.00905·7-s + 1.36·8-s − 1.67·9-s + 0.988·10-s + 0.627·11-s + 0.310·12-s + 0.117·13-s + 0.00653·14-s + 0.326·15-s − 0.0899·16-s − 2.14·17-s + 1.21·18-s − 3.31·19-s + 1.78·20-s + 0.00215·21-s − 0.453·22-s + 0.568·23-s − 0.324·24-s + 0.581·25-s − 0.0847·26-s + 0.816·27-s + 0.0118·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(100169.\)
Root analytic conductor: \(4.21785\)
Motivic weight: \(53\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 1,\ (\ :53/2, 53/2, 53/2, 53/2),\ 1)\)

Particular Values

\(L(27)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{55}{2})\) 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)$
good2$C_2 \wr S_4$ \( 1 + 2139885 p^{5} T + 1003667114255 p^{14} T^{2} + 22859181486091845 p^{25} T^{3} + 78860968145917904739 p^{41} T^{4} + 22859181486091845 p^{78} T^{5} + 1003667114255 p^{120} T^{6} + 2139885 p^{164} T^{7} + p^{212} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 116490111920 p^{2} T + \)\(51\!\cdots\!40\)\( p^{8} T^{2} - \)\(76\!\cdots\!60\)\( p^{16} T^{3} + \)\(27\!\cdots\!02\)\( p^{26} T^{4} - \)\(76\!\cdots\!60\)\( p^{69} T^{5} + \)\(51\!\cdots\!40\)\( p^{114} T^{6} + 116490111920 p^{161} T^{7} + p^{212} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 36511166346354504 p^{3} T + \)\(36\!\cdots\!52\)\( p^{8} T^{2} + \)\(32\!\cdots\!12\)\( p^{14} T^{3} + \)\(21\!\cdots\!74\)\( p^{21} T^{4} + \)\(32\!\cdots\!12\)\( p^{67} T^{5} + \)\(36\!\cdots\!52\)\( p^{114} T^{6} + 36511166346354504 p^{162} T^{7} + p^{212} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 32120252964063632800 p T + \)\(50\!\cdots\!00\)\( p^{4} T^{2} + \)\(57\!\cdots\!00\)\( p^{8} T^{3} + \)\(10\!\cdots\!14\)\( p^{13} T^{4} + \)\(57\!\cdots\!00\)\( p^{61} T^{5} + \)\(50\!\cdots\!00\)\( p^{110} T^{6} + 32120252964063632800 p^{160} T^{7} + p^{212} T^{8} \)
11$C_2 \wr S_4$ \( 1 - \)\(22\!\cdots\!48\)\( p T + \)\(37\!\cdots\!08\)\( p^{2} T^{2} - \)\(57\!\cdots\!76\)\( p^{5} T^{3} + \)\(40\!\cdots\!70\)\( p^{9} T^{4} - \)\(57\!\cdots\!76\)\( p^{58} T^{5} + \)\(37\!\cdots\!08\)\( p^{108} T^{6} - \)\(22\!\cdots\!48\)\( p^{160} T^{7} + p^{212} T^{8} \)
13$C_2 \wr S_4$ \( 1 - \)\(29\!\cdots\!80\)\( p T + \)\(55\!\cdots\!80\)\( p^{4} T^{2} + \)\(40\!\cdots\!60\)\( p^{5} T^{3} + \)\(19\!\cdots\!54\)\( p^{7} T^{4} + \)\(40\!\cdots\!60\)\( p^{58} T^{5} + \)\(55\!\cdots\!80\)\( p^{110} T^{6} - \)\(29\!\cdots\!80\)\( p^{160} T^{7} + p^{212} T^{8} \)
17$C_2 \wr S_4$ \( 1 + \)\(51\!\cdots\!80\)\( p T + \)\(10\!\cdots\!20\)\( p^{3} T^{2} + \)\(14\!\cdots\!60\)\( p^{5} T^{3} + \)\(21\!\cdots\!06\)\( p^{7} T^{4} + \)\(14\!\cdots\!60\)\( p^{58} T^{5} + \)\(10\!\cdots\!20\)\( p^{109} T^{6} + \)\(51\!\cdots\!80\)\( p^{160} T^{7} + p^{212} T^{8} \)
19$C_2 \wr S_4$ \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(19\!\cdots\!44\)\( p T^{2} + \)\(10\!\cdots\!20\)\( p^{2} T^{3} + \)\(23\!\cdots\!66\)\( p^{4} T^{4} + \)\(10\!\cdots\!20\)\( p^{55} T^{5} + \)\(19\!\cdots\!44\)\( p^{107} T^{6} + \)\(25\!\cdots\!60\)\( p^{159} T^{7} + p^{212} T^{8} \)
23$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!20\)\( p T + \)\(95\!\cdots\!80\)\( p^{2} T^{2} - \)\(98\!\cdots\!80\)\( p^{4} T^{3} + \)\(73\!\cdots\!02\)\( p^{6} T^{4} - \)\(98\!\cdots\!80\)\( p^{57} T^{5} + \)\(95\!\cdots\!80\)\( p^{108} T^{6} - \)\(30\!\cdots\!20\)\( p^{160} T^{7} + p^{212} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(14\!\cdots\!40\)\( T + \)\(64\!\cdots\!64\)\( p T^{2} + \)\(17\!\cdots\!80\)\( p^{2} T^{3} + \)\(41\!\cdots\!34\)\( p^{3} T^{4} + \)\(17\!\cdots\!80\)\( p^{55} T^{5} + \)\(64\!\cdots\!64\)\( p^{107} T^{6} + \)\(14\!\cdots\!40\)\( p^{159} T^{7} + p^{212} T^{8} \)
31$C_2 \wr S_4$ \( 1 + \)\(39\!\cdots\!32\)\( T + \)\(33\!\cdots\!48\)\( T^{2} + \)\(32\!\cdots\!64\)\( p T^{3} + \)\(55\!\cdots\!70\)\( p^{2} T^{4} + \)\(32\!\cdots\!64\)\( p^{54} T^{5} + \)\(33\!\cdots\!48\)\( p^{106} T^{6} + \)\(39\!\cdots\!32\)\( p^{159} T^{7} + p^{212} T^{8} \)
37$C_2 \wr S_4$ \( 1 + \)\(28\!\cdots\!80\)\( T + \)\(97\!\cdots\!40\)\( p T^{2} + \)\(88\!\cdots\!40\)\( p^{2} T^{3} + \)\(11\!\cdots\!06\)\( p^{3} T^{4} + \)\(88\!\cdots\!40\)\( p^{55} T^{5} + \)\(97\!\cdots\!40\)\( p^{107} T^{6} + \)\(28\!\cdots\!80\)\( p^{159} T^{7} + p^{212} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(19\!\cdots\!68\)\( p T + \)\(40\!\cdots\!48\)\( p^{2} T^{2} + \)\(12\!\cdots\!84\)\( p^{3} T^{3} + \)\(79\!\cdots\!70\)\( p^{4} T^{4} + \)\(12\!\cdots\!84\)\( p^{56} T^{5} + \)\(40\!\cdots\!48\)\( p^{108} T^{6} - \)\(19\!\cdots\!68\)\( p^{160} T^{7} + p^{212} T^{8} \)
43$C_2 \wr S_4$ \( 1 - \)\(45\!\cdots\!00\)\( T + \)\(51\!\cdots\!00\)\( p T^{2} - \)\(30\!\cdots\!00\)\( p^{2} T^{3} + \)\(17\!\cdots\!14\)\( p^{3} T^{4} - \)\(30\!\cdots\!00\)\( p^{55} T^{5} + \)\(51\!\cdots\!00\)\( p^{107} T^{6} - \)\(45\!\cdots\!00\)\( p^{159} T^{7} + p^{212} T^{8} \)
47$C_2 \wr S_4$ \( 1 + \)\(95\!\cdots\!20\)\( p T + \)\(98\!\cdots\!60\)\( p^{2} T^{2} + \)\(55\!\cdots\!60\)\( p^{3} T^{3} + \)\(30\!\cdots\!18\)\( p^{4} T^{4} + \)\(55\!\cdots\!60\)\( p^{56} T^{5} + \)\(98\!\cdots\!60\)\( p^{108} T^{6} + \)\(95\!\cdots\!20\)\( p^{160} T^{7} + p^{212} T^{8} \)
53$C_2 \wr S_4$ \( 1 - \)\(38\!\cdots\!20\)\( T + \)\(13\!\cdots\!80\)\( p T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!60\)\( p^{53} T^{5} + \)\(13\!\cdots\!80\)\( p^{107} T^{6} - \)\(38\!\cdots\!20\)\( p^{159} T^{7} + p^{212} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(27\!\cdots\!80\)\( p T + \)\(23\!\cdots\!16\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!40\)\( p^{53} T^{5} + \)\(23\!\cdots\!16\)\( p^{106} T^{6} - \)\(27\!\cdots\!80\)\( p^{160} T^{7} + p^{212} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(65\!\cdots\!28\)\( T + \)\(79\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!76\)\( p^{53} T^{5} + \)\(79\!\cdots\!68\)\( p^{106} T^{6} - \)\(65\!\cdots\!28\)\( p^{159} T^{7} + p^{212} T^{8} \)
67$C_2 \wr S_4$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(16\!\cdots\!60\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(27\!\cdots\!20\)\( p^{53} T^{5} + \)\(16\!\cdots\!60\)\( p^{106} T^{6} + \)\(11\!\cdots\!60\)\( p^{159} T^{7} + p^{212} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(37\!\cdots\!08\)\( T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!96\)\( p^{53} T^{5} + \)\(37\!\cdots\!08\)\( p^{106} T^{6} - \)\(35\!\cdots\!48\)\( p^{159} T^{7} + p^{212} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(70\!\cdots\!40\)\( T + \)\(40\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!20\)\( p^{53} T^{5} + \)\(40\!\cdots\!20\)\( p^{106} T^{6} + \)\(70\!\cdots\!40\)\( p^{159} T^{7} + p^{212} T^{8} \)
79$C_2 \wr S_4$ \( 1 + \)\(15\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!26\)\( T^{4} + \)\(11\!\cdots\!80\)\( p^{53} T^{5} + \)\(10\!\cdots\!56\)\( p^{106} T^{6} + \)\(15\!\cdots\!40\)\( p^{159} T^{7} + p^{212} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(41\!\cdots\!60\)\( T^{2} + \)\(45\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!38\)\( T^{4} + \)\(45\!\cdots\!60\)\( p^{53} T^{5} + \)\(41\!\cdots\!60\)\( p^{106} T^{6} + \)\(26\!\cdots\!20\)\( p^{159} T^{7} + p^{212} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(37\!\cdots\!20\)\( T + \)\(51\!\cdots\!76\)\( T^{2} + \)\(97\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} + \)\(97\!\cdots\!40\)\( p^{53} T^{5} + \)\(51\!\cdots\!76\)\( p^{106} T^{6} + \)\(37\!\cdots\!20\)\( p^{159} T^{7} + p^{212} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(42\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{53} T^{5} + \)\(42\!\cdots\!40\)\( p^{106} T^{6} - \)\(10\!\cdots\!60\)\( p^{159} T^{7} + p^{212} T^{8} \)
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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

−14.36551689731113482282739137020, −13.09996580144090454415565086225, −13.03731209265050034915417118205, −12.90052040179762709374073777583, −11.60301157721301318006491683168, −11.41715500917187292499776701244, −11.02693080545101520574031167217, −10.68358842611485570456741833126, −9.727609437338406029370805436637, −8.982004386372900206391170042899, −8.833179282512613733816156938047, −8.518912255990319196151859506115, −8.327112005454431297516498750343, −7.22583473262764177733905317129, −7.04990757063851416267034010110, −6.08708573159931613225512413659, −5.89849610083662869023730756189, −4.86517752815356537152155760419, −4.54848879576851445901379130219, −4.05411492846685872317216066274, −3.86475122900726968429150086588, −3.10089646196511199445192809885, −2.12491545208015807400443058476, −2.08200369414158559766490705768, −1.11128581718251547380353913934, 0, 0, 0, 0, 1.11128581718251547380353913934, 2.08200369414158559766490705768, 2.12491545208015807400443058476, 3.10089646196511199445192809885, 3.86475122900726968429150086588, 4.05411492846685872317216066274, 4.54848879576851445901379130219, 4.86517752815356537152155760419, 5.89849610083662869023730756189, 6.08708573159931613225512413659, 7.04990757063851416267034010110, 7.22583473262764177733905317129, 8.327112005454431297516498750343, 8.518912255990319196151859506115, 8.833179282512613733816156938047, 8.982004386372900206391170042899, 9.727609437338406029370805436637, 10.68358842611485570456741833126, 11.02693080545101520574031167217, 11.41715500917187292499776701244, 11.60301157721301318006491683168, 12.90052040179762709374073777583, 13.03731209265050034915417118205, 13.09996580144090454415565086225, 14.36551689731113482282739137020

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