Properties

Label 16-12e24-1.1-c2e8-0-2
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 6·7-s + 36·11-s − 14·13-s − 4·19-s + 102·23-s − 27·25-s − 114·29-s − 50·31-s − 36·35-s − 120·37-s − 264·41-s + 28·43-s − 150·47-s + 163·49-s − 216·55-s − 108·59-s − 14·61-s + 84·65-s + 20·67-s − 76·73-s + 216·77-s + 26·79-s + 246·83-s − 84·91-s + 24·95-s − 236·97-s + ⋯
L(s)  = 1  − 6/5·5-s + 6/7·7-s + 3.27·11-s − 1.07·13-s − 0.210·19-s + 4.43·23-s − 1.07·25-s − 3.93·29-s − 1.61·31-s − 1.02·35-s − 3.24·37-s − 6.43·41-s + 0.651·43-s − 3.19·47-s + 3.32·49-s − 3.92·55-s − 1.83·59-s − 0.229·61-s + 1.29·65-s + 0.298·67-s − 1.04·73-s + 2.80·77-s + 0.329·79-s + 2.96·83-s − 0.923·91-s + 0.252·95-s − 2.43·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1114287611\)
\(L(\frac12)\) \(\approx\) \(0.1114287611\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 6 T + 63 T^{2} + 306 T^{3} + 2461 T^{4} + 15132 T^{5} + 86562 T^{6} + 473928 T^{7} + 2131866 T^{8} + 473928 p^{2} T^{9} + 86562 p^{4} T^{10} + 15132 p^{6} T^{11} + 2461 p^{8} T^{12} + 306 p^{10} T^{13} + 63 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \)
7 \( 1 - 6 T - 127 T^{2} + 102 p T^{3} + 1459 p T^{4} - 43992 T^{5} - 588922 T^{6} + 917052 T^{7} + 31082074 T^{8} + 917052 p^{2} T^{9} - 588922 p^{4} T^{10} - 43992 p^{6} T^{11} + 1459 p^{9} T^{12} + 102 p^{11} T^{13} - 127 p^{12} T^{14} - 6 p^{14} T^{15} + p^{16} T^{16} \)
11 \( 1 - 36 T + 828 T^{2} - 1296 p T^{3} + 189103 T^{4} - 184500 p T^{5} + 18127476 T^{6} - 142796448 T^{7} + 1322450688 T^{8} - 142796448 p^{2} T^{9} + 18127476 p^{4} T^{10} - 184500 p^{7} T^{11} + 189103 p^{8} T^{12} - 1296 p^{11} T^{13} + 828 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \)
13 \( 1 + 14 T - 129 T^{2} - 1886 T^{3} + 9461 T^{4} + 158556 T^{5} + 4694986 T^{6} + 8508992 T^{7} - 1078024230 T^{8} + 8508992 p^{2} T^{9} + 4694986 p^{4} T^{10} + 158556 p^{6} T^{11} + 9461 p^{8} T^{12} - 1886 p^{10} T^{13} - 129 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} \)
17 \( 1 - 858 T^{2} + 505105 T^{4} - 204328674 T^{6} + 68217206628 T^{8} - 204328674 p^{4} T^{10} + 505105 p^{8} T^{12} - 858 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 2 T + 265 T^{2} + 6170 T^{3} + 157036 T^{4} + 6170 p^{2} T^{5} + 265 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 102 T + 6033 T^{2} - 261630 T^{3} + 9057109 T^{4} - 271036392 T^{5} + 7331036166 T^{6} - 184354538700 T^{7} + 4374045926298 T^{8} - 184354538700 p^{2} T^{9} + 7331036166 p^{4} T^{10} - 271036392 p^{6} T^{11} + 9057109 p^{8} T^{12} - 261630 p^{10} T^{13} + 6033 p^{12} T^{14} - 102 p^{14} T^{15} + p^{16} T^{16} \)
29 \( 1 + 114 T + 8599 T^{2} + 486438 T^{3} + 22645933 T^{4} + 890300916 T^{5} + 31365917458 T^{6} + 1000841966952 T^{7} + 29952505780714 T^{8} + 1000841966952 p^{2} T^{9} + 31365917458 p^{4} T^{10} + 890300916 p^{6} T^{11} + 22645933 p^{8} T^{12} + 486438 p^{10} T^{13} + 8599 p^{12} T^{14} + 114 p^{14} T^{15} + p^{16} T^{16} \)
31 \( 1 + 50 T - 21 p T^{2} - 68990 T^{3} - 334663 T^{4} + 5769840 T^{5} - 1286987822 T^{6} + 24925573940 T^{7} + 3245120663634 T^{8} + 24925573940 p^{2} T^{9} - 1286987822 p^{4} T^{10} + 5769840 p^{6} T^{11} - 334663 p^{8} T^{12} - 68990 p^{10} T^{13} - 21 p^{13} T^{14} + 50 p^{14} T^{15} + p^{16} T^{16} \)
37 \( ( 1 + 60 T + 5200 T^{2} + 219060 T^{3} + 10695774 T^{4} + 219060 p^{2} T^{5} + 5200 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 + 264 T + 38158 T^{2} + 3940464 T^{3} + 321724129 T^{4} + 21802117680 T^{5} + 1260648967822 T^{6} + 63190156979928 T^{7} + 2768191141578052 T^{8} + 63190156979928 p^{2} T^{9} + 1260648967822 p^{4} T^{10} + 21802117680 p^{6} T^{11} + 321724129 p^{8} T^{12} + 3940464 p^{10} T^{13} + 38158 p^{12} T^{14} + 264 p^{14} T^{15} + p^{16} T^{16} \)
43 \( 1 - 28 T - 2364 T^{2} - 141848 T^{3} + 7021127 T^{4} + 338054892 T^{5} + 12175270060 T^{6} - 693178978408 T^{7} - 23735268373680 T^{8} - 693178978408 p^{2} T^{9} + 12175270060 p^{4} T^{10} + 338054892 p^{6} T^{11} + 7021127 p^{8} T^{12} - 141848 p^{10} T^{13} - 2364 p^{12} T^{14} - 28 p^{14} T^{15} + p^{16} T^{16} \)
47 \( 1 + 150 T + 16249 T^{2} + 1312350 T^{3} + 88432117 T^{4} + 5053131000 T^{5} + 262267521478 T^{6} + 12644089287900 T^{7} + 595883042253514 T^{8} + 12644089287900 p^{2} T^{9} + 262267521478 p^{4} T^{10} + 5053131000 p^{6} T^{11} + 88432117 p^{8} T^{12} + 1312350 p^{10} T^{13} + 16249 p^{12} T^{14} + 150 p^{14} T^{15} + p^{16} T^{16} \)
53 \( 1 - 15456 T^{2} + 115082620 T^{4} - 547509829536 T^{6} + 1820499555493830 T^{8} - 547509829536 p^{4} T^{10} + 115082620 p^{8} T^{12} - 15456 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 + 108 T + 14820 T^{2} + 1180656 T^{3} + 104177575 T^{4} + 7489417356 T^{5} + 509063599692 T^{6} + 32920352322696 T^{7} + 1903340903262096 T^{8} + 32920352322696 p^{2} T^{9} + 509063599692 p^{4} T^{10} + 7489417356 p^{6} T^{11} + 104177575 p^{8} T^{12} + 1180656 p^{10} T^{13} + 14820 p^{12} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16} \)
61 \( 1 + 14 T - 6633 T^{2} - 472862 T^{3} + 19891733 T^{4} + 2361270204 T^{5} + 99631636810 T^{6} - 5929775489824 T^{7} - 599620944879702 T^{8} - 5929775489824 p^{2} T^{9} + 99631636810 p^{4} T^{10} + 2361270204 p^{6} T^{11} + 19891733 p^{8} T^{12} - 472862 p^{10} T^{13} - 6633 p^{12} T^{14} + 14 p^{14} T^{15} + p^{16} T^{16} \)
67 \( 1 - 20 T - 15756 T^{2} + 141560 T^{3} + 151491335 T^{4} - 601531260 T^{5} - 1010166803204 T^{6} + 1012785207880 T^{7} + 5163639663893904 T^{8} + 1012785207880 p^{2} T^{9} - 1010166803204 p^{4} T^{10} - 601531260 p^{6} T^{11} + 151491335 p^{8} T^{12} + 141560 p^{10} T^{13} - 15756 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} \)
71 \( 1 - 13464 T^{2} + 105773404 T^{4} - 696180849192 T^{6} + 3915811319568198 T^{8} - 696180849192 p^{4} T^{10} + 105773404 p^{8} T^{12} - 13464 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 38 T + 11485 T^{2} + 120314 T^{3} + 68572624 T^{4} + 120314 p^{2} T^{5} + 11485 p^{4} T^{6} + 38 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 26 T - 12687 T^{2} + 773558 T^{3} + 64925093 T^{4} - 5836272936 T^{5} - 59707396922 T^{6} + 18106848362500 T^{7} - 289214375790438 T^{8} + 18106848362500 p^{2} T^{9} - 59707396922 p^{4} T^{10} - 5836272936 p^{6} T^{11} + 64925093 p^{8} T^{12} + 773558 p^{10} T^{13} - 12687 p^{12} T^{14} - 26 p^{14} T^{15} + p^{16} T^{16} \)
83 \( 1 - 246 T + 54825 T^{2} - 8524638 T^{3} + 1254860197 T^{4} - 148166306376 T^{5} + 16481417300694 T^{6} - 1562002778781228 T^{7} + 140096195439334362 T^{8} - 1562002778781228 p^{2} T^{9} + 16481417300694 p^{4} T^{10} - 148166306376 p^{6} T^{11} + 1254860197 p^{8} T^{12} - 8524638 p^{10} T^{13} + 54825 p^{12} T^{14} - 246 p^{14} T^{15} + p^{16} T^{16} \)
89 \( 1 - 28448 T^{2} + 504887356 T^{4} - 6266477720288 T^{6} + 56631012864364294 T^{8} - 6266477720288 p^{4} T^{10} + 504887356 p^{8} T^{12} - 28448 p^{12} T^{14} + p^{16} T^{16} \)
97 \( 1 + 236 T + 438 T^{2} - 1185512 T^{3} + 510091505 T^{4} + 51879290952 T^{5} - 3191784386426 T^{6} + 104812309462772 T^{7} + 86158688483213604 T^{8} + 104812309462772 p^{2} T^{9} - 3191784386426 p^{4} T^{10} + 51879290952 p^{6} T^{11} + 510091505 p^{8} T^{12} - 1185512 p^{10} T^{13} + 438 p^{12} T^{14} + 236 p^{14} T^{15} + p^{16} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.60114990608278234483707343456, −3.58684635544399245700670122144, −3.54452178019266909428412921854, −3.45045265044303906298965590567, −3.36117843499160567250211420311, −3.26684246145157382369430499570, −3.16457504587726492181395272120, −2.75232082987585170758711505074, −2.70926049682672661201191273370, −2.61094275921199283692786714639, −2.36624881950427896347338328540, −2.34096427472597273601036494649, −1.98025779127961336006936070659, −1.70645928753495736245691805821, −1.69540286260085326358642933717, −1.68020613837888585908528889249, −1.53562694455455529686181960881, −1.50088489684653561499663480315, −1.46489717876542075751430039768, −1.29823693942041866154520807330, −0.69834515623225336019608680188, −0.55433940509330858525728798044, −0.55290336676679586601001058861, −0.31246093977749412606350861851, −0.02664613178696710304341314327, 0.02664613178696710304341314327, 0.31246093977749412606350861851, 0.55290336676679586601001058861, 0.55433940509330858525728798044, 0.69834515623225336019608680188, 1.29823693942041866154520807330, 1.46489717876542075751430039768, 1.50088489684653561499663480315, 1.53562694455455529686181960881, 1.68020613837888585908528889249, 1.69540286260085326358642933717, 1.70645928753495736245691805821, 1.98025779127961336006936070659, 2.34096427472597273601036494649, 2.36624881950427896347338328540, 2.61094275921199283692786714639, 2.70926049682672661201191273370, 2.75232082987585170758711505074, 3.16457504587726492181395272120, 3.26684246145157382369430499570, 3.36117843499160567250211420311, 3.45045265044303906298965590567, 3.54452178019266909428412921854, 3.58684635544399245700670122144, 3.60114990608278234483707343456

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

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