Properties

Degree 8
Conductor $ 1 $
Sign $1$
Motivic weight 55
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.08e8·2-s − 6.82e12·3-s − 3.09e16·4-s + 1.43e19·5-s − 1.42e21·6-s − 2.05e23·7-s − 1.04e25·8-s − 3.01e26·9-s + 3.00e27·10-s + 1.94e28·11-s + 2.11e29·12-s − 4.44e30·13-s − 4.28e31·14-s − 9.82e31·15-s + 8.55e30·16-s + 8.59e33·17-s − 6.29e34·18-s + 3.39e35·19-s − 4.45e35·20-s + 1.40e36·21-s + 4.06e36·22-s + 4.98e37·23-s + 7.14e37·24-s − 4.33e38·25-s − 9.28e38·26-s − 2.83e38·27-s + 6.35e39·28-s + ⋯
L(s)  = 1  + 1.09·2-s − 0.516·3-s − 0.858·4-s + 0.864·5-s − 0.567·6-s − 1.18·7-s − 1.53·8-s − 1.72·9-s + 0.949·10-s + 0.448·11-s + 0.443·12-s − 1.03·13-s − 1.29·14-s − 0.446·15-s + 0.00659·16-s + 1.24·17-s − 1.90·18-s + 2.31·19-s − 0.741·20-s + 0.610·21-s + 0.492·22-s + 1.77·23-s + 0.790·24-s − 1.56·25-s − 1.13·26-s − 0.122·27-s + 1.01·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(56-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(55\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 1,\ (\ :55/2, 55/2, 55/2, 55/2),\ 1)$
$L(28)$  $\approx$  $1.56258$
$L(\frac12)$  $\approx$  $1.56258$
$L(\frac{57}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 8.
$p$$\Gal(F_p)$$F_p$
good2$C_2 \wr S_4$ \( 1 - 26077815 p^{3} T + 72710344310675 p^{10} T^{2} - 5491911371338614105 p^{21} T^{3} + \)\(18\!\cdots\!59\)\( p^{37} T^{4} - 5491911371338614105 p^{76} T^{5} + 72710344310675 p^{120} T^{6} - 26077815 p^{168} T^{7} + p^{220} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 252647062640 p^{3} T + \)\(58\!\cdots\!00\)\( p^{10} T^{2} + \)\(40\!\cdots\!80\)\( p^{19} T^{3} + \)\(30\!\cdots\!02\)\( p^{30} T^{4} + \)\(40\!\cdots\!80\)\( p^{74} T^{5} + \)\(58\!\cdots\!00\)\( p^{120} T^{6} + 252647062640 p^{168} T^{7} + p^{220} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 2879387592677433432 p T + \)\(20\!\cdots\!56\)\( p^{5} T^{2} - \)\(10\!\cdots\!16\)\( p^{10} T^{3} + \)\(31\!\cdots\!66\)\( p^{17} T^{4} - \)\(10\!\cdots\!16\)\( p^{65} T^{5} + \)\(20\!\cdots\!56\)\( p^{115} T^{6} - 2879387592677433432 p^{166} T^{7} + p^{220} T^{8} \)
7$C_2 \wr S_4$ \( 1 + \)\(59\!\cdots\!00\)\( p^{3} T + \)\(67\!\cdots\!00\)\( p^{7} T^{2} + \)\(15\!\cdots\!00\)\( p^{12} T^{3} + \)\(38\!\cdots\!02\)\( p^{18} T^{4} + \)\(15\!\cdots\!00\)\( p^{67} T^{5} + \)\(67\!\cdots\!00\)\( p^{117} T^{6} + \)\(59\!\cdots\!00\)\( p^{168} T^{7} + p^{220} T^{8} \)
11$C_2 \wr S_4$ \( 1 - \)\(17\!\cdots\!28\)\( p T + \)\(18\!\cdots\!88\)\( p^{3} T^{2} + \)\(56\!\cdots\!44\)\( p^{5} T^{3} + \)\(45\!\cdots\!70\)\( p^{8} T^{4} + \)\(56\!\cdots\!44\)\( p^{60} T^{5} + \)\(18\!\cdots\!88\)\( p^{113} T^{6} - \)\(17\!\cdots\!28\)\( p^{166} T^{7} + p^{220} T^{8} \)
13$C_2 \wr S_4$ \( 1 + \)\(34\!\cdots\!20\)\( p T + \)\(25\!\cdots\!00\)\( p^{3} T^{2} + \)\(40\!\cdots\!80\)\( p^{6} T^{3} + \)\(13\!\cdots\!26\)\( p^{9} T^{4} + \)\(40\!\cdots\!80\)\( p^{61} T^{5} + \)\(25\!\cdots\!00\)\( p^{113} T^{6} + \)\(34\!\cdots\!20\)\( p^{166} T^{7} + p^{220} T^{8} \)
17$C_2 \wr S_4$ \( 1 - \)\(50\!\cdots\!80\)\( p T + \)\(65\!\cdots\!00\)\( p^{2} T^{2} - \)\(13\!\cdots\!80\)\( p^{4} T^{3} + \)\(32\!\cdots\!26\)\( p^{7} T^{4} - \)\(13\!\cdots\!80\)\( p^{59} T^{5} + \)\(65\!\cdots\!00\)\( p^{112} T^{6} - \)\(50\!\cdots\!80\)\( p^{166} T^{7} + p^{220} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(64\!\cdots\!84\)\( p T^{2} - \)\(33\!\cdots\!00\)\( p^{3} T^{3} + \)\(17\!\cdots\!94\)\( p^{5} T^{4} - \)\(33\!\cdots\!00\)\( p^{58} T^{5} + \)\(64\!\cdots\!84\)\( p^{111} T^{6} - \)\(33\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
23$C_2 \wr S_4$ \( 1 - \)\(49\!\cdots\!60\)\( T + \)\(10\!\cdots\!00\)\( p T^{2} - \)\(73\!\cdots\!60\)\( p^{3} T^{3} + \)\(42\!\cdots\!86\)\( p^{5} T^{4} - \)\(73\!\cdots\!60\)\( p^{58} T^{5} + \)\(10\!\cdots\!00\)\( p^{111} T^{6} - \)\(49\!\cdots\!60\)\( p^{165} T^{7} + p^{220} T^{8} \)
29$C_2 \wr S_4$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(63\!\cdots\!96\)\( T^{2} + \)\(32\!\cdots\!00\)\( p T^{3} + \)\(30\!\cdots\!66\)\( p^{2} T^{4} + \)\(32\!\cdots\!00\)\( p^{56} T^{5} + \)\(63\!\cdots\!96\)\( p^{110} T^{6} + \)\(18\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
31$C_2 \wr S_4$ \( 1 - \)\(22\!\cdots\!08\)\( T + \)\(18\!\cdots\!88\)\( p T^{2} - \)\(75\!\cdots\!96\)\( p^{2} T^{3} + \)\(32\!\cdots\!70\)\( p^{3} T^{4} - \)\(75\!\cdots\!96\)\( p^{57} T^{5} + \)\(18\!\cdots\!88\)\( p^{111} T^{6} - \)\(22\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
37$C_2 \wr S_4$ \( 1 + \)\(32\!\cdots\!20\)\( T + \)\(22\!\cdots\!00\)\( p T^{2} + \)\(98\!\cdots\!40\)\( p^{2} T^{3} + \)\(41\!\cdots\!66\)\( p^{3} T^{4} + \)\(98\!\cdots\!40\)\( p^{57} T^{5} + \)\(22\!\cdots\!00\)\( p^{111} T^{6} + \)\(32\!\cdots\!20\)\( p^{165} T^{7} + p^{220} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(24\!\cdots\!08\)\( T + \)\(46\!\cdots\!88\)\( p^{2} T^{2} - \)\(12\!\cdots\!76\)\( p^{2} T^{3} + \)\(90\!\cdots\!70\)\( p^{3} T^{4} - \)\(12\!\cdots\!76\)\( p^{57} T^{5} + \)\(46\!\cdots\!88\)\( p^{112} T^{6} - \)\(24\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(20\!\cdots\!00\)\( p T + \)\(14\!\cdots\!00\)\( p^{2} T^{2} + \)\(20\!\cdots\!00\)\( p^{3} T^{3} + \)\(80\!\cdots\!98\)\( p^{4} T^{4} + \)\(20\!\cdots\!00\)\( p^{58} T^{5} + \)\(14\!\cdots\!00\)\( p^{112} T^{6} + \)\(20\!\cdots\!00\)\( p^{166} T^{7} + p^{220} T^{8} \)
47$C_2 \wr S_4$ \( 1 - \)\(90\!\cdots\!20\)\( p T + \)\(27\!\cdots\!00\)\( p^{2} T^{2} + \)\(79\!\cdots\!60\)\( p^{3} T^{3} - \)\(14\!\cdots\!42\)\( p^{4} T^{4} + \)\(79\!\cdots\!60\)\( p^{58} T^{5} + \)\(27\!\cdots\!00\)\( p^{112} T^{6} - \)\(90\!\cdots\!20\)\( p^{166} T^{7} + p^{220} T^{8} \)
53$C_2 \wr S_4$ \( 1 + \)\(92\!\cdots\!60\)\( p T + \)\(11\!\cdots\!00\)\( p^{2} T^{2} + \)\(66\!\cdots\!80\)\( p^{3} T^{3} + \)\(41\!\cdots\!58\)\( p^{4} T^{4} + \)\(66\!\cdots\!80\)\( p^{58} T^{5} + \)\(11\!\cdots\!00\)\( p^{112} T^{6} + \)\(92\!\cdots\!60\)\( p^{166} T^{7} + p^{220} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(81\!\cdots\!00\)\( T + \)\(88\!\cdots\!96\)\( T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!06\)\( T^{4} - \)\(42\!\cdots\!00\)\( p^{55} T^{5} + \)\(88\!\cdots\!96\)\( p^{110} T^{6} - \)\(81\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(70\!\cdots\!08\)\( T + \)\(37\!\cdots\!28\)\( T^{2} - \)\(39\!\cdots\!56\)\( T^{3} + \)\(67\!\cdots\!70\)\( T^{4} - \)\(39\!\cdots\!56\)\( p^{55} T^{5} + \)\(37\!\cdots\!28\)\( p^{110} T^{6} - \)\(70\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(41\!\cdots\!60\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(36\!\cdots\!80\)\( T^{3} + \)\(74\!\cdots\!98\)\( T^{4} - \)\(36\!\cdots\!80\)\( p^{55} T^{5} + \)\(16\!\cdots\!00\)\( p^{110} T^{6} - \)\(41\!\cdots\!60\)\( p^{165} T^{7} + p^{220} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(99\!\cdots\!08\)\( T + \)\(16\!\cdots\!28\)\( T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + \)\(83\!\cdots\!70\)\( T^{4} - \)\(51\!\cdots\!56\)\( p^{55} T^{5} + \)\(16\!\cdots\!28\)\( p^{110} T^{6} - \)\(99\!\cdots\!08\)\( p^{165} T^{7} + p^{220} T^{8} \)
73$C_2 \wr S_4$ \( 1 + \)\(89\!\cdots\!40\)\( T + \)\(85\!\cdots\!00\)\( T^{2} + \)\(99\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + \)\(99\!\cdots\!80\)\( p^{55} T^{5} + \)\(85\!\cdots\!00\)\( p^{110} T^{6} + \)\(89\!\cdots\!40\)\( p^{165} T^{7} + p^{220} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(48\!\cdots\!00\)\( T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!06\)\( T^{4} - \)\(35\!\cdots\!00\)\( p^{55} T^{5} + \)\(16\!\cdots\!96\)\( p^{110} T^{6} - \)\(48\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
83$C_2 \wr S_4$ \( 1 + \)\(71\!\cdots\!20\)\( T + \)\(88\!\cdots\!00\)\( T^{2} + \)\(64\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(64\!\cdots\!40\)\( p^{55} T^{5} + \)\(88\!\cdots\!00\)\( p^{110} T^{6} + \)\(71\!\cdots\!20\)\( p^{165} T^{7} + p^{220} T^{8} \)
89$C_2 \wr S_4$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(13\!\cdots\!96\)\( T^{2} - \)\(81\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!06\)\( T^{4} - \)\(81\!\cdots\!00\)\( p^{55} T^{5} + \)\(13\!\cdots\!96\)\( p^{110} T^{6} - \)\(15\!\cdots\!00\)\( p^{165} T^{7} + p^{220} T^{8} \)
97$C_2 \wr S_4$ \( 1 - \)\(51\!\cdots\!40\)\( T + \)\(69\!\cdots\!00\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!98\)\( T^{4} - \)\(28\!\cdots\!20\)\( p^{55} T^{5} + \)\(69\!\cdots\!00\)\( p^{110} T^{6} - \)\(51\!\cdots\!40\)\( p^{165} T^{7} + p^{220} T^{8} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.95118231308152375064742191944, −12.20664586550508961397681257035, −11.87075427096932695407937365486, −11.61267383697861039540824713270, −10.81170957746528457970535173058, −9.846693394851963566904888415547, −9.671107198526613643329808404749, −9.438438160703926861457700939837, −8.964199280923750619958448691525, −7.929393418680684726188677607875, −7.84000730608031809753268719439, −6.69359478163537418462844897959, −6.48228350335189907889910527584, −5.68326504619685570106718169026, −5.32510389581082330361180733567, −5.29216002292109809460747630907, −4.84686041653671905108901001592, −3.81047115268815694039929272110, −3.44084370216387245829347446565, −3.26013001793659364410915219565, −2.62468760178042997638962466692, −2.04453281596194447419305485636, −1.15076563335679149777486733227, −0.67796267284066736667574685265, −0.25509747719462604680015866627, 0.25509747719462604680015866627, 0.67796267284066736667574685265, 1.15076563335679149777486733227, 2.04453281596194447419305485636, 2.62468760178042997638962466692, 3.26013001793659364410915219565, 3.44084370216387245829347446565, 3.81047115268815694039929272110, 4.84686041653671905108901001592, 5.29216002292109809460747630907, 5.32510389581082330361180733567, 5.68326504619685570106718169026, 6.48228350335189907889910527584, 6.69359478163537418462844897959, 7.84000730608031809753268719439, 7.929393418680684726188677607875, 8.964199280923750619958448691525, 9.438438160703926861457700939837, 9.671107198526613643329808404749, 9.846693394851963566904888415547, 10.81170957746528457970535173058, 11.61267383697861039540824713270, 11.87075427096932695407937365486, 12.20664586550508961397681257035, 12.95118231308152375064742191944

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