Properties

Degree 16
Conductor $ 2^{16} \cdot 3^{24} $
Sign $1$
Motivic weight 4
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s + 13·7-s + 18·11-s − 5·13-s + 562·19-s + 1.71e3·23-s − 1.03e3·25-s − 2.11e3·29-s + 187·31-s + 117·35-s + 16·37-s + 7.92e3·41-s − 68·43-s − 1.36e4·47-s + 4.72e3·49-s + 162·55-s + 2.00e4·59-s − 1.93e3·61-s − 45·65-s + 154·67-s − 7.80e3·73-s + 234·77-s − 2.19e3·79-s − 3.70e4·83-s − 65·91-s + 5.05e3·95-s + 7.28e3·97-s + ⋯
L(s)  = 1  + 9/25·5-s + 0.265·7-s + 0.148·11-s − 0.0295·13-s + 1.55·19-s + 3.24·23-s − 1.65·25-s − 2.51·29-s + 0.194·31-s + 0.0955·35-s + 0.0116·37-s + 4.71·41-s − 0.0367·43-s − 6.19·47-s + 1.96·49-s + 0.0535·55-s + 5.76·59-s − 0.520·61-s − 0.0106·65-s + 0.0343·67-s − 1.46·73-s + 0.0394·77-s − 0.351·79-s − 5.37·83-s − 0.00784·91-s + 0.560·95-s + 0.773·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{16} \cdot 3^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{16} \cdot 3^{24} ,\ ( \ : [2]^{8} ),\ 1 )\)
\(L(\frac{5}{2})\)  \(\approx\)  \(5.70064\)
\(L(\frac12)\)  \(\approx\)  \(5.70064\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 9 T + 1114 T^{2} - 9783 T^{3} + 533599 T^{4} - 10528056 T^{5} + 59806456 T^{6} - 10305069192 T^{7} - 6001445444 T^{8} - 10305069192 p^{4} T^{9} + 59806456 p^{8} T^{10} - 10528056 p^{12} T^{11} + 533599 p^{16} T^{12} - 9783 p^{20} T^{13} + 1114 p^{24} T^{14} - 9 p^{28} T^{15} + p^{32} T^{16} \)
7 \( 1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 7093476 p^{2} T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 447642835484 p^{4} T^{9} + 2901735784 p^{8} T^{10} - 7093476 p^{14} T^{11} + 8962391 p^{16} T^{12} + 124753 p^{20} T^{13} - 4554 p^{24} T^{14} - 13 p^{28} T^{15} + p^{32} T^{16} \)
11 \( 1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 445095012639474 p^{4} T^{9} + 9074477968558 p^{8} T^{10} - 17443950048 p^{12} T^{11} + 452208721 p^{16} T^{12} - 607356 p^{20} T^{13} + 33850 p^{24} T^{14} - 18 p^{28} T^{15} + p^{32} T^{16} \)
13 \( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 3464858176098460 p^{4} T^{9} + 23191943061904 p^{8} T^{10} + 250092029040 p^{12} T^{11} + 352176575 p^{16} T^{12} - 7266665 p^{20} T^{13} - 42054 p^{24} T^{14} + 5 p^{28} T^{15} + p^{32} T^{16} \)
17 \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - 6759733382202755 p^{8} T^{10} + 52320681154 p^{16} T^{12} - 288125 p^{24} T^{14} + p^{32} T^{16} \)
19 \( ( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 68843041 p^{4} T^{5} + 110170 p^{8} T^{6} - 281 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
23 \( 1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} + \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!68\)\( p^{4} T^{9} - 118516645434237428 p^{8} T^{10} + 16333135609500 p^{12} T^{11} + 342642958747 p^{16} T^{12} - 935945649 p^{20} T^{13} + 1529458 p^{24} T^{14} - 1719 p^{28} T^{15} + p^{32} T^{16} \)
29 \( 1 + 2115 T + 4091014 T^{2} + 5498870985 T^{3} + 7048695081595 T^{4} + 8024737206821040 T^{5} + 8297140249856169556 T^{6} + \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} + \)\(79\!\cdots\!80\)\( p^{4} T^{9} + 8297140249856169556 p^{8} T^{10} + 8024737206821040 p^{12} T^{11} + 7048695081595 p^{16} T^{12} + 5498870985 p^{20} T^{13} + 4091014 p^{24} T^{14} + 2115 p^{28} T^{15} + p^{32} T^{16} \)
31 \( 1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 9550301616 p T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} - 13487262715981377034 p^{4} T^{9} - 3460282108678916846 p^{8} T^{10} - 9550301616 p^{13} T^{11} + 3362431719041 p^{16} T^{12} + 186847537 p^{20} T^{13} - 2516004 p^{24} T^{14} - 187 p^{28} T^{15} + p^{32} T^{16} \)
37 \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 1256575624 p^{4} T^{5} + 3611368 p^{8} T^{6} - 8 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
41 \( 1 - 7920 T + 37687894 T^{2} - 132890424480 T^{3} + 385083705354505 T^{4} - 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} - \)\(41\!\cdots\!20\)\( p^{4} T^{9} + \)\(21\!\cdots\!66\)\( p^{8} T^{10} - 963185727644706960 p^{12} T^{11} + 385083705354505 p^{16} T^{12} - 132890424480 p^{20} T^{13} + 37687894 p^{24} T^{14} - 7920 p^{28} T^{15} + p^{32} T^{16} \)
43 \( 1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} + \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} + \)\(27\!\cdots\!16\)\( p^{4} T^{9} - \)\(54\!\cdots\!36\)\( p^{8} T^{10} + 59145034173804 p^{12} T^{11} + 102574547445791 p^{16} T^{12} - 209786648 p^{20} T^{13} - 12950604 p^{24} T^{14} + 68 p^{28} T^{15} + p^{32} T^{16} \)
47 \( 1 + 13689 T + 103685338 T^{2} + 12006252297 p T^{3} + 2435028217967227 T^{4} + 185888262507277500 p T^{5} + \)\(26\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!56\)\( p T^{7} + \)\(76\!\cdots\!80\)\( p^{2} T^{8} + \)\(15\!\cdots\!56\)\( p^{5} T^{9} + \)\(26\!\cdots\!72\)\( p^{8} T^{10} + 185888262507277500 p^{13} T^{11} + 2435028217967227 p^{16} T^{12} + 12006252297 p^{21} T^{13} + 103685338 p^{24} T^{14} + 13689 p^{28} T^{15} + p^{32} T^{16} \)
53 \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - 84051566001475463360 p^{8} T^{10} + 115452291970684 p^{16} T^{12} - 5145920 p^{24} T^{14} + p^{32} T^{16} \)
59 \( 1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} - \)\(84\!\cdots\!76\)\( p^{4} T^{9} + \)\(21\!\cdots\!48\)\( p^{8} T^{10} - 49579528565018409012 p^{12} T^{11} + 9931594296358591 p^{16} T^{12} - 1657982214864 p^{20} T^{13} + 216711700 p^{24} T^{14} - 20052 p^{28} T^{15} + p^{32} T^{16} \)
61 \( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(40\!\cdots\!44\)\( p^{4} T^{9} + \)\(12\!\cdots\!24\)\( p^{8} T^{10} - 1350680282380662864 p^{12} T^{11} + 416288373490931 p^{16} T^{12} + 149647181023 p^{20} T^{13} - 10529634 p^{24} T^{14} + 1937 p^{28} T^{15} + p^{32} T^{16} \)
67 \( 1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} - \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} - \)\(11\!\cdots\!78\)\( p^{4} T^{9} + \)\(70\!\cdots\!02\)\( p^{8} T^{10} + 738160924156362336 p^{12} T^{11} + 539365905411977 p^{16} T^{12} - 25606229228 p^{20} T^{13} - 33835854 p^{24} T^{14} - 154 p^{28} T^{15} + p^{32} T^{16} \)
71 \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(11\!\cdots\!24\)\( p^{8} T^{10} + 3244147638477940 p^{16} T^{12} - 68871716 p^{24} T^{14} + p^{32} T^{16} \)
73 \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 292589317519 p^{4} T^{5} + 59309470 p^{8} T^{6} + 3901 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
79 \( 1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} + \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} + \)\(76\!\cdots\!80\)\( p^{4} T^{9} - \)\(16\!\cdots\!36\)\( p^{8} T^{10} + 3711455091635884260 p^{12} T^{11} + 3128319215246375 p^{16} T^{12} - 187644610415 p^{20} T^{13} - 87724914 p^{24} T^{14} + 2195 p^{28} T^{15} + p^{32} T^{16} \)
83 \( 1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} + \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} + \)\(40\!\cdots\!76\)\( p^{4} T^{9} + \)\(62\!\cdots\!28\)\( p^{8} T^{10} + \)\(87\!\cdots\!72\)\( p^{12} T^{11} + 104510585134438411 p^{16} T^{12} + 9956481997959 p^{20} T^{13} + 725723290 p^{24} T^{14} + 37017 p^{28} T^{15} + p^{32} T^{16} \)
89 \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(48\!\cdots\!04\)\( p^{8} T^{10} + 46567064448316540 p^{16} T^{12} - 294759296 p^{24} T^{14} + p^{32} T^{16} \)
97 \( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(33\!\cdots\!26\)\( p^{4} T^{9} - \)\(75\!\cdots\!16\)\( p^{8} T^{10} - 97460573991801682656 p^{12} T^{11} + 57034963146137471 p^{16} T^{12} + 993163976152 p^{20} T^{13} - 283226964 p^{24} T^{14} - 7282 p^{28} T^{15} + p^{32} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.61060217594513426481135999985, −5.48849351409636477732708277135, −5.23363382367211402946438630424, −5.10856116051280890647854153359, −4.96462948044494793577214220625, −4.72265259848176459687293811234, −4.45089109264747375366187411470, −4.25605076254616409835985134904, −4.17445744714903414580955188443, −3.93077098241084770941989653096, −3.69375349289803353284616410463, −3.60889813924494450626032073028, −3.19587413054919385278757209766, −3.13900122944042677893112737325, −2.77971099993704726223552790299, −2.69831798993300103991175674647, −2.57642796736828470645343009869, −2.02686155251008694596792019030, −1.96526593375826607732971837729, −1.66508312469006032719636975083, −1.28004882657192918321811050719, −1.11538410422801554956599198626, −0.949928618562229170525563108687, −0.47976083246582776993304898284, −0.25181078929944249186948372453, 0.25181078929944249186948372453, 0.47976083246582776993304898284, 0.949928618562229170525563108687, 1.11538410422801554956599198626, 1.28004882657192918321811050719, 1.66508312469006032719636975083, 1.96526593375826607732971837729, 2.02686155251008694596792019030, 2.57642796736828470645343009869, 2.69831798993300103991175674647, 2.77971099993704726223552790299, 3.13900122944042677893112737325, 3.19587413054919385278757209766, 3.60889813924494450626032073028, 3.69375349289803353284616410463, 3.93077098241084770941989653096, 4.17445744714903414580955188443, 4.25605076254616409835985134904, 4.45089109264747375366187411470, 4.72265259848176459687293811234, 4.96462948044494793577214220625, 5.10856116051280890647854153359, 5.23363382367211402946438630424, 5.48849351409636477732708277135, 5.61060217594513426481135999985

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

Plot not available for L-functions of degree greater than 10.