Properties

Degree 6
Conductor $ 2^{3} $
Sign $1$
Motivic weight 49
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 5.03e7·2-s − 1.62e10·3-s + 1.68e15·4-s − 1.01e17·5-s − 8.15e17·6-s − 1.25e20·7-s + 4.72e22·8-s − 1.64e23·9-s − 5.12e24·10-s − 2.58e25·11-s − 2.73e25·12-s + 2.94e27·13-s − 6.31e27·14-s + 1.64e27·15-s + 1.18e30·16-s + 3.91e30·17-s − 8.25e30·18-s + 1.45e31·19-s − 1.71e32·20-s + 2.03e30·21-s − 1.30e33·22-s − 4.95e33·23-s − 7.65e32·24-s − 1.73e33·25-s + 1.48e35·26-s + 6.93e34·27-s − 2.11e35·28-s + ⋯
L(s)  = 1  + 2.12·2-s − 0.0331·3-s + 3·4-s − 0.763·5-s − 0.0702·6-s − 0.247·7-s + 3.53·8-s − 0.685·9-s − 1.62·10-s − 0.791·11-s − 0.0993·12-s + 1.50·13-s − 0.525·14-s + 0.0253·15-s + 15/4·16-s + 2.79·17-s − 1.45·18-s + 0.683·19-s − 2.29·20-s + 0.00819·21-s − 1.67·22-s − 2.15·23-s − 0.117·24-s − 0.0977·25-s + 3.19·26-s + 0.592·27-s − 0.742·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+49/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(49\)
character  :  induced by $\chi_{2} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 8,\ (\ :49/2, 49/2, 49/2),\ 1)$
$L(25)$  $\approx$  $9.458362593$
$L(\frac12)$  $\approx$  $9.458362593$
$L(\frac{51}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 6. If $p = 2$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - p^{24} T )^{3} \)
good3$S_4\times C_2$ \( 1 + 5401204796 p T + 75154918447963115531 p^{7} T^{2} - \)\(55\!\cdots\!68\)\( p^{19} T^{3} + 75154918447963115531 p^{56} T^{4} + 5401204796 p^{99} T^{5} + p^{147} T^{6} \)
5$S_4\times C_2$ \( 1 + 20362603280368086 p T + \)\(38\!\cdots\!59\)\( p^{5} T^{2} + \)\(93\!\cdots\!56\)\( p^{13} T^{3} + \)\(38\!\cdots\!59\)\( p^{54} T^{4} + 20362603280368086 p^{99} T^{5} + p^{147} T^{6} \)
7$S_4\times C_2$ \( 1 + 2560799576126608104 p^{2} T + \)\(12\!\cdots\!99\)\( p^{5} T^{2} - \)\(30\!\cdots\!84\)\( p^{9} T^{3} + \)\(12\!\cdots\!99\)\( p^{54} T^{4} + 2560799576126608104 p^{100} T^{5} + p^{147} T^{6} \)
11$S_4\times C_2$ \( 1 + \)\(25\!\cdots\!04\)\( T + \)\(72\!\cdots\!95\)\( p^{3} T^{2} - \)\(18\!\cdots\!40\)\( p^{7} T^{3} + \)\(72\!\cdots\!95\)\( p^{52} T^{4} + \)\(25\!\cdots\!04\)\( p^{98} T^{5} + p^{147} T^{6} \)
13$S_4\times C_2$ \( 1 - \)\(29\!\cdots\!42\)\( T + \)\(10\!\cdots\!39\)\( p T^{2} - \)\(78\!\cdots\!16\)\( p^{4} T^{3} + \)\(10\!\cdots\!39\)\( p^{50} T^{4} - \)\(29\!\cdots\!42\)\( p^{98} T^{5} + p^{147} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(39\!\cdots\!54\)\( T + \)\(41\!\cdots\!39\)\( p T^{2} - \)\(19\!\cdots\!16\)\( p^{3} T^{3} + \)\(41\!\cdots\!39\)\( p^{50} T^{4} - \)\(39\!\cdots\!54\)\( p^{98} T^{5} + p^{147} T^{6} \)
19$S_4\times C_2$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(65\!\cdots\!23\)\( p T^{2} - \)\(18\!\cdots\!60\)\( p^{3} T^{3} + \)\(65\!\cdots\!23\)\( p^{50} T^{4} - \)\(14\!\cdots\!80\)\( p^{98} T^{5} + p^{147} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(21\!\cdots\!96\)\( p T + \)\(19\!\cdots\!31\)\( p^{3} T^{2} + \)\(86\!\cdots\!48\)\( p^{5} T^{3} + \)\(19\!\cdots\!31\)\( p^{52} T^{4} + \)\(21\!\cdots\!96\)\( p^{99} T^{5} + p^{147} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(93\!\cdots\!30\)\( T + \)\(32\!\cdots\!83\)\( p T^{2} - \)\(10\!\cdots\!40\)\( p^{2} T^{3} + \)\(32\!\cdots\!83\)\( p^{50} T^{4} - \)\(93\!\cdots\!30\)\( p^{98} T^{5} + p^{147} T^{6} \)
31$S_4\times C_2$ \( 1 - \)\(24\!\cdots\!56\)\( T + \)\(54\!\cdots\!75\)\( p T^{2} - \)\(74\!\cdots\!60\)\( p^{2} T^{3} + \)\(54\!\cdots\!75\)\( p^{50} T^{4} - \)\(24\!\cdots\!56\)\( p^{98} T^{5} + p^{147} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(30\!\cdots\!74\)\( T + \)\(57\!\cdots\!79\)\( p T^{2} - \)\(29\!\cdots\!32\)\( p^{2} T^{3} + \)\(57\!\cdots\!79\)\( p^{50} T^{4} - \)\(30\!\cdots\!74\)\( p^{98} T^{5} + p^{147} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!54\)\( T + \)\(69\!\cdots\!55\)\( p T^{2} + \)\(14\!\cdots\!20\)\( p^{2} T^{3} + \)\(69\!\cdots\!55\)\( p^{50} T^{4} + \)\(11\!\cdots\!54\)\( p^{98} T^{5} + p^{147} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(17\!\cdots\!48\)\( T + \)\(75\!\cdots\!79\)\( p T^{2} + \)\(19\!\cdots\!76\)\( p^{2} T^{3} + \)\(75\!\cdots\!79\)\( p^{50} T^{4} + \)\(17\!\cdots\!48\)\( p^{98} T^{5} + p^{147} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(46\!\cdots\!48\)\( p T + \)\(15\!\cdots\!57\)\( p^{2} T^{2} + \)\(33\!\cdots\!44\)\( p^{3} T^{3} + \)\(15\!\cdots\!57\)\( p^{51} T^{4} + \)\(46\!\cdots\!48\)\( p^{99} T^{5} + p^{147} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(74\!\cdots\!02\)\( T + \)\(61\!\cdots\!67\)\( T^{2} - \)\(55\!\cdots\!36\)\( T^{3} + \)\(61\!\cdots\!67\)\( p^{49} T^{4} - \)\(74\!\cdots\!02\)\( p^{98} T^{5} + p^{147} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(89\!\cdots\!60\)\( p T + \)\(22\!\cdots\!17\)\( T^{2} + \)\(63\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!17\)\( p^{49} T^{4} + \)\(89\!\cdots\!60\)\( p^{99} T^{5} + p^{147} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(12\!\cdots\!94\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!35\)\( p^{49} T^{4} + \)\(12\!\cdots\!94\)\( p^{98} T^{5} + p^{147} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!24\)\( T + \)\(13\!\cdots\!33\)\( T^{2} - \)\(71\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!33\)\( p^{49} T^{4} - \)\(11\!\cdots\!24\)\( p^{98} T^{5} + p^{147} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(40\!\cdots\!24\)\( T + \)\(18\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!85\)\( p^{49} T^{4} + \)\(40\!\cdots\!24\)\( p^{98} T^{5} + p^{147} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(83\!\cdots\!47\)\( p^{49} T^{4} + \)\(10\!\cdots\!18\)\( p^{98} T^{5} + p^{147} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(33\!\cdots\!57\)\( T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!57\)\( p^{49} T^{4} - \)\(36\!\cdots\!60\)\( p^{98} T^{5} + p^{147} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(15\!\cdots\!92\)\( T + \)\(23\!\cdots\!97\)\( T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(23\!\cdots\!97\)\( p^{49} T^{4} - \)\(15\!\cdots\!92\)\( p^{98} T^{5} + p^{147} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(54\!\cdots\!70\)\( T + \)\(86\!\cdots\!27\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!27\)\( p^{49} T^{4} - \)\(54\!\cdots\!70\)\( p^{98} T^{5} + p^{147} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(43\!\cdots\!14\)\( T + \)\(35\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(35\!\cdots\!83\)\( p^{49} T^{4} - \)\(43\!\cdots\!14\)\( p^{98} T^{5} + p^{147} T^{6} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.63842697430683460631080616061, −13.75322535842257672037216602098, −13.69632637739654228276171914503, −12.94587927038859581927827316841, −12.16424367654486949592547232511, −11.78308669141420651321467337635, −11.78024926452517289789238985645, −10.71265707610207407597099757239, −10.28255603028618506949661735125, −9.687572339335342823233636254144, −8.188336359086953354755005804921, −8.029601410615503291712199546654, −7.66940722106239554714971244108, −6.51203288617538966076572647104, −6.14815329312843700095546581612, −5.78881220888448067201822684551, −5.05986559254949980047816221721, −4.66930669446981506766037510014, −3.87923887090072069028369432781, −3.25043929444362311762323158344, −3.24898992284991690458622643024, −2.65988708517971336327308455761, −1.48282513455735403202035086127, −1.35713215212954564094521439019, −0.37534543783491370376268612734, 0.37534543783491370376268612734, 1.35713215212954564094521439019, 1.48282513455735403202035086127, 2.65988708517971336327308455761, 3.24898992284991690458622643024, 3.25043929444362311762323158344, 3.87923887090072069028369432781, 4.66930669446981506766037510014, 5.05986559254949980047816221721, 5.78881220888448067201822684551, 6.14815329312843700095546581612, 6.51203288617538966076572647104, 7.66940722106239554714971244108, 8.029601410615503291712199546654, 8.188336359086953354755005804921, 9.687572339335342823233636254144, 10.28255603028618506949661735125, 10.71265707610207407597099757239, 11.78024926452517289789238985645, 11.78308669141420651321467337635, 12.16424367654486949592547232511, 12.94587927038859581927827316841, 13.69632637739654228276171914503, 13.75322535842257672037216602098, 14.63842697430683460631080616061

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