Properties

Label 8-1-1.1-c47e4-0-0
Degree $8$
Conductor $1$
Sign $1$
Analytic cond. $38314.7$
Root an. cond. $3.74042$
Motivic weight $47$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.78e6·2-s + 3.84e10·3-s − 6.23e13·4-s − 3.11e16·5-s + 2.22e17·6-s − 3.91e19·7-s + 1.00e20·8-s − 6.09e22·9-s − 1.80e23·10-s − 1.90e24·11-s − 2.39e24·12-s + 1.26e26·13-s − 2.26e26·14-s − 1.19e27·15-s + 2.17e28·16-s + 2.10e29·17-s − 3.52e29·18-s − 1.05e30·19-s + 1.93e30·20-s − 1.50e30·21-s − 1.10e31·22-s + 1.37e32·23-s + 3.88e30·24-s − 4.32e32·25-s + 7.34e32·26-s − 5.15e33·27-s + 2.44e33·28-s + ⋯
L(s)  = 1  + 0.487·2-s + 0.235·3-s − 0.442·4-s − 1.16·5-s + 0.115·6-s − 0.540·7-s + 0.0604·8-s − 2.29·9-s − 0.569·10-s − 0.640·11-s − 0.104·12-s + 0.843·13-s − 0.263·14-s − 0.275·15-s + 1.09·16-s + 2.55·17-s − 1.11·18-s − 0.942·19-s + 0.517·20-s − 0.127·21-s − 0.312·22-s + 1.37·23-s + 0.0142·24-s − 0.608·25-s + 0.411·26-s − 1.18·27-s + 0.239·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(38314.7\)
Root analytic conductor: \(3.74042\)
Motivic weight: \(47\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1,\ (\ :47/2, 47/2, 47/2, 47/2),\ 1)\)

Particular Values

\(L(24)\) \(\approx\) \(1.626991539\)
\(L(\frac12)\) \(\approx\) \(1.626991539\)
\(L(\frac{49}{2})\) 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)$
good2$C_2 \wr S_4$ \( 1 - 723195 p^{3} T + 93562089245 p^{10} T^{2} - 242210623551345 p^{22} T^{3} - 68020972333510731 p^{37} T^{4} - 242210623551345 p^{69} T^{5} + 93562089245 p^{104} T^{6} - 723195 p^{144} T^{7} + p^{188} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 4273499440 p^{2} T + 28556588122484284580 p^{7} T^{2} + \)\(94\!\cdots\!80\)\( p^{16} T^{3} + \)\(26\!\cdots\!74\)\( p^{27} T^{4} + \)\(94\!\cdots\!80\)\( p^{63} T^{5} + 28556588122484284580 p^{101} T^{6} - 4273499440 p^{143} T^{7} + p^{188} T^{8} \)
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

−14.21310863853175597063482089839, −13.36363814561436126495704941728, −13.11442815552733184083047136647, −12.31445719615043186233989073150, −11.94815917211133557240383180920, −11.28244498175268667193628020180, −11.03054489242947403771681137383, −10.07922282019894674193145975946, −9.767291605655760643534174749608, −8.742156910971676986389001236815, −8.415174341120272693876340904649, −8.026379096700647627059786266112, −7.59247495885116737306597296132, −6.75012880687945371681079481690, −5.86172135153587139218720329263, −5.40281269628152395213217208803, −5.40006152176154725533900714111, −4.25474528483862484526695563349, −3.64203140782896131177921063585, −3.40086324840605483838717490068, −2.99795514297973204710885199718, −2.34778698908042292893419555331, −1.37250117030602044036451903916, −0.68178057301413172356960766406, −0.32132544407402366206956330567, 0.32132544407402366206956330567, 0.68178057301413172356960766406, 1.37250117030602044036451903916, 2.34778698908042292893419555331, 2.99795514297973204710885199718, 3.40086324840605483838717490068, 3.64203140782896131177921063585, 4.25474528483862484526695563349, 5.40006152176154725533900714111, 5.40281269628152395213217208803, 5.86172135153587139218720329263, 6.75012880687945371681079481690, 7.59247495885116737306597296132, 8.026379096700647627059786266112, 8.415174341120272693876340904649, 8.742156910971676986389001236815, 9.767291605655760643534174749608, 10.07922282019894674193145975946, 11.03054489242947403771681137383, 11.28244498175268667193628020180, 11.94815917211133557240383180920, 12.31445719615043186233989073150, 13.11442815552733184083047136647, 13.36363814561436126495704941728, 14.21310863853175597063482089839

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