↑

Properties

Label 8-162e4-1.1-c11e4-0-6
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $2.40038\times 10^{8}$
Root an. cond. $11.1566$
Motivic weight $11$
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  + 64·2-s + 1.02e3·4-s − 6.90e3·5-s − 8.05e3·7-s − 6.55e4·8-s − 4.41e5·10-s − 1.84e5·11-s + 4.51e4·13-s − 5.15e5·14-s − 4.19e6·16-s − 6.72e6·17-s − 6.60e6·19-s − 7.06e6·20-s − 1.17e7·22-s + 1.68e7·23-s + 6.27e7·25-s + 2.89e6·26-s − 8.24e6·28-s − 1.73e8·29-s + 1.91e7·31-s − 6.71e7·32-s − 4.30e8·34-s + 5.55e7·35-s + 7.76e8·37-s − 4.22e8·38-s + 4.52e8·40-s + 1.88e8·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.987·5-s − 0.181·7-s − 0.707·8-s − 1.39·10-s − 0.344·11-s + 0.0337·13-s − 0.256·14-s − 16-s − 1.14·17-s − 0.611·19-s − 0.493·20-s − 0.487·22-s + 0.544·23-s + 1.28·25-s + 0.0477·26-s − 0.0905·28-s − 1.57·29-s + 0.119·31-s − 0.353·32-s − 1.62·34-s + 0.178·35-s + 1.84·37-s − 0.865·38-s + 0.698·40-s + 0.254·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.40038\times 10^{8}\)
Root analytic conductor: \(11.1566\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(14.33237779\)
\(L(\frac12)\) \(\approx\) \(14.33237779\)
\(L(\frac{13}{2})\) 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)$
bad2$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 + 276 p^{2} T - 604069 p^{2} T^{2} - 385787556 p^{4} T^{3} - 1241811791664 p^{4} T^{4} - 385787556 p^{15} T^{5} - 604069 p^{24} T^{6} + 276 p^{35} T^{7} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 + 1150 p T - 2914768562 T^{2} - 1121344787600 p T^{3} + 4839886948661562295 T^{4} - 1121344787600 p^{12} T^{5} - 2914768562 p^{22} T^{6} + 1150 p^{34} T^{7} + p^{44} T^{8} \)
11$D_4\times C_2$ \( 1 + 184194 T - 401396753746 T^{2} - 2265572098107360 p T^{3} + \)\(81\!\cdots\!67\)\( p^{2} T^{4} - 2265572098107360 p^{12} T^{5} - 401396753746 p^{22} T^{6} + 184194 p^{33} T^{7} + p^{44} T^{8} \)
13$D_4\times C_2$ \( 1 - 45176 T - 3547619262353 T^{2} + 1565829738760120 T^{3} + \)\(93\!\cdots\!60\)\( T^{4} + 1565829738760120 p^{11} T^{5} - 3547619262353 p^{22} T^{6} - 45176 p^{33} T^{7} + p^{44} T^{8} \)
17$D_{4}$ \( ( 1 + 3362466 T + 63904765129459 T^{2} + 3362466 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 3302342 T + 144175065402078 T^{2} + 3302342 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 16811226 T - 1506905004098938 T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(19\!\cdots\!40\)\( p^{11} T^{5} - 1506905004098938 p^{22} T^{6} - 16811226 p^{33} T^{7} + p^{44} T^{8} \)
29$D_4\times C_2$ \( 1 + 173467800 T + 8201135592373967 T^{2} - \)\(43\!\cdots\!00\)\( T^{3} - \)\(36\!\cdots\!52\)\( T^{4} - \)\(43\!\cdots\!00\)\( p^{11} T^{5} + 8201135592373967 p^{22} T^{6} + 173467800 p^{33} T^{7} + p^{44} T^{8} \)
31$D_4\times C_2$ \( 1 - 19117064 T - 19591233150682190 T^{2} + \)\(58\!\cdots\!64\)\( T^{3} - \)\(25\!\cdots\!41\)\( T^{4} + \)\(58\!\cdots\!64\)\( p^{11} T^{5} - 19591233150682190 p^{22} T^{6} - 19117064 p^{33} T^{7} + p^{44} T^{8} \)
37$D_{4}$ \( ( 1 - 388355920 T + 352750138351354425 T^{2} - 388355920 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 188837760 T - 9849909461223518 p T^{2} + \)\(12\!\cdots\!40\)\( T^{3} - \)\(12\!\cdots\!37\)\( T^{4} + \)\(12\!\cdots\!40\)\( p^{11} T^{5} - 9849909461223518 p^{23} T^{6} - 188837760 p^{33} T^{7} + p^{44} T^{8} \)
43$D_4\times C_2$ \( 1 + 1449050698 T + 137522053848747550 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!39\)\( T^{4} + \)\(15\!\cdots\!20\)\( p^{11} T^{5} + 137522053848747550 p^{22} T^{6} + 1449050698 p^{33} T^{7} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 + 851790372 T - 4175152557303119374 T^{2} - \)\(37\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(37\!\cdots\!56\)\( p^{11} T^{5} - 4175152557303119374 p^{22} T^{6} + 851790372 p^{33} T^{7} + p^{44} T^{8} \)
53$D_{4}$ \( ( 1 - 3837253404 T + 21463090483683068062 T^{2} - 3837253404 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 1293665796 T + 13841008348409834378 T^{2} - \)\(93\!\cdots\!80\)\( T^{3} - \)\(81\!\cdots\!81\)\( T^{4} - \)\(93\!\cdots\!80\)\( p^{11} T^{5} + 13841008348409834378 p^{22} T^{6} + 1293665796 p^{33} T^{7} + p^{44} T^{8} \)
61$D_4\times C_2$ \( 1 - 8437713704 T - 25713993172936767521 T^{2} - \)\(83\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!72\)\( T^{4} - \)\(83\!\cdots\!60\)\( p^{11} T^{5} - 25713993172936767521 p^{22} T^{6} - 8437713704 p^{33} T^{7} + p^{44} T^{8} \)
67$D_4\times C_2$ \( 1 - 3349274150 T - \)\(18\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!00\)\( p^{11} T^{5} - \)\(18\!\cdots\!22\)\( p^{22} T^{6} - 3349274150 p^{33} T^{7} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 - 24518665194 T + \)\(48\!\cdots\!70\)\( T^{2} - 24518665194 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 25410413174 T + \)\(37\!\cdots\!47\)\( T^{2} + 25410413174 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 43134100886 T - 89023851268900572650 T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(18\!\cdots\!91\)\( T^{4} - \)\(19\!\cdots\!68\)\( p^{11} T^{5} - 89023851268900572650 p^{22} T^{6} - 43134100886 p^{33} T^{7} + p^{44} T^{8} \)
83$D_4\times C_2$ \( 1 - 15242519964 T - \)\(22\!\cdots\!86\)\( T^{2} + \)\(13\!\cdots\!28\)\( T^{3} + \)\(42\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!28\)\( p^{11} T^{5} - \)\(22\!\cdots\!86\)\( p^{22} T^{6} - 15242519964 p^{33} T^{7} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 - 109553463366 T + \)\(70\!\cdots\!31\)\( T^{2} - 109553463366 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 155454832136 T + \)\(51\!\cdots\!42\)\( T^{2} - \)\(73\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!19\)\( T^{4} - \)\(73\!\cdots\!28\)\( p^{11} T^{5} + \)\(51\!\cdots\!42\)\( p^{22} T^{6} - 155454832136 p^{33} T^{7} + p^{44} T^{8} \)
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

−7.40857516654083211355487098958, −6.97447088458025329692467154596, −6.64619954048986500835959603003, −6.51681716125499028820691461040, −6.24490347067316179677537167167, −5.92931285091753391620402295987, −5.57071065189801500092408883877, −5.19297450517715219034795853075, −5.15474928356173482393355451883, −4.84531796086362726244140964812, −4.28687399657135137679437484179, −4.25467622255922726785085159998, −4.21251193494302733421192860349, −3.61486445364132908016401911771, −3.45443370716785837010669259555, −3.18776157228734944299235323476, −2.95140470576162586667883759401, −2.40227429188538125849974353736, −2.18306312843223152425929477010, −1.96366213532102259826046429987, −1.74225620676847571429976921404, −0.72316439981409803412411858366, −0.68568810738264316946848920696, −0.65195103555284004722682309039, −0.41396101986801345788296164340, 0.41396101986801345788296164340, 0.65195103555284004722682309039, 0.68568810738264316946848920696, 0.72316439981409803412411858366, 1.74225620676847571429976921404, 1.96366213532102259826046429987, 2.18306312843223152425929477010, 2.40227429188538125849974353736, 2.95140470576162586667883759401, 3.18776157228734944299235323476, 3.45443370716785837010669259555, 3.61486445364132908016401911771, 4.21251193494302733421192860349, 4.25467622255922726785085159998, 4.28687399657135137679437484179, 4.84531796086362726244140964812, 5.15474928356173482393355451883, 5.19297450517715219034795853075, 5.57071065189801500092408883877, 5.92931285091753391620402295987, 6.24490347067316179677537167167, 6.51681716125499028820691461040, 6.64619954048986500835959603003, 6.97447088458025329692467154596, 7.40857516654083211355487098958

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