Properties

Label 32-20e16-1.1-c8e16-0-0
Degree $32$
Conductor $6.554\times 10^{20}$
Sign $1$
Analytic cond. $3.77095\times 10^{14}$
Root an. cond. $2.85439$
Motivic weight $8$
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  + 6·2-s − 8·4-s − 4.84e3·8-s + 3.30e4·9-s + 5.13e4·13-s − 4.88e4·16-s + 2.75e4·17-s + 1.98e5·18-s + 6.25e5·25-s + 3.08e5·26-s + 2.76e6·29-s − 9.59e5·32-s + 1.65e5·34-s − 2.64e5·36-s + 9.00e6·37-s − 8.57e6·41-s + 2.97e7·49-s + 3.75e6·50-s − 4.11e5·52-s + 2.45e6·53-s + 1.65e7·58-s + 8.37e6·61-s − 1.27e6·64-s − 2.20e5·68-s − 1.60e8·72-s + 6.19e7·73-s + 5.40e7·74-s + ⋯
L(s)  = 1  + 3/8·2-s − 0.0312·4-s − 1.18·8-s + 5.04·9-s + 1.79·13-s − 0.744·16-s + 0.329·17-s + 1.89·18-s + 8/5·25-s + 0.674·26-s + 3.90·29-s − 0.914·32-s + 0.123·34-s − 0.157·36-s + 4.80·37-s − 3.03·41-s + 5.15·49-s + 3/5·50-s − 0.0562·52-s + 0.310·53-s + 1.46·58-s + 0.604·61-s − 0.0757·64-s − 0.0103·68-s − 5.96·72-s + 2.17·73-s + 1.80·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(3.77095\times 10^{14}\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 5^{16} ,\ ( \ : [4]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(186.5527907\)
\(L(\frac12)\) \(\approx\) \(186.5527907\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 3 p T + 11 p^{2} T^{2} + 567 p^{3} T^{3} - 447 p^{4} T^{4} + 7491 p^{7} T^{5} + 13519 p^{10} T^{6} - 6999 p^{13} T^{7} + 96437 p^{16} T^{8} - 6999 p^{21} T^{9} + 13519 p^{26} T^{10} + 7491 p^{31} T^{11} - 447 p^{36} T^{12} + 567 p^{43} T^{13} + 11 p^{50} T^{14} - 3 p^{57} T^{15} + p^{64} T^{16} \)
5 \( ( 1 - p^{7} T^{2} )^{8} \)
good3 \( 1 - 33088 T^{2} + 575902904 T^{4} - 2384637859648 p T^{6} + 889715980752604 p^{4} T^{8} - 23380858584547080640 p^{3} T^{10} + \)\(62\!\cdots\!72\)\( p^{4} T^{12} - \)\(56\!\cdots\!68\)\( p^{8} T^{14} + \)\(47\!\cdots\!18\)\( p^{12} T^{16} - \)\(56\!\cdots\!68\)\( p^{24} T^{18} + \)\(62\!\cdots\!72\)\( p^{36} T^{20} - 23380858584547080640 p^{51} T^{22} + 889715980752604 p^{68} T^{24} - 2384637859648 p^{81} T^{26} + 575902904 p^{96} T^{28} - 33088 p^{112} T^{30} + p^{128} T^{32} \)
7 \( 1 - 29716608 T^{2} + 549029353332024 T^{4} - \)\(15\!\cdots\!76\)\( p^{2} T^{6} + \)\(34\!\cdots\!64\)\( p^{4} T^{8} - \)\(66\!\cdots\!20\)\( p^{6} T^{10} + \)\(10\!\cdots\!12\)\( p^{8} T^{12} - \)\(15\!\cdots\!12\)\( p^{10} T^{14} + \)\(19\!\cdots\!98\)\( p^{12} T^{16} - \)\(15\!\cdots\!12\)\( p^{26} T^{18} + \)\(10\!\cdots\!12\)\( p^{40} T^{20} - \)\(66\!\cdots\!20\)\( p^{54} T^{22} + \)\(34\!\cdots\!64\)\( p^{68} T^{24} - \)\(15\!\cdots\!76\)\( p^{82} T^{26} + 549029353332024 p^{96} T^{28} - 29716608 p^{112} T^{30} + p^{128} T^{32} \)
11 \( 1 - 1595449296 T^{2} + 109557327527852520 p T^{4} - \)\(52\!\cdots\!60\)\( p T^{6} + \)\(18\!\cdots\!20\)\( p T^{8} - \)\(52\!\cdots\!68\)\( T^{10} + \)\(98\!\cdots\!68\)\( p T^{12} - \)\(19\!\cdots\!40\)\( T^{14} + \)\(38\!\cdots\!70\)\( T^{16} - \)\(19\!\cdots\!40\)\( p^{16} T^{18} + \)\(98\!\cdots\!68\)\( p^{33} T^{20} - \)\(52\!\cdots\!68\)\( p^{48} T^{22} + \)\(18\!\cdots\!20\)\( p^{65} T^{24} - \)\(52\!\cdots\!60\)\( p^{81} T^{26} + 109557327527852520 p^{97} T^{28} - 1595449296 p^{112} T^{30} + p^{128} T^{32} \)
13 \( ( 1 - 25696 T + 2606936824 T^{2} - 81475212513824 T^{3} + 5003239356127079708 T^{4} - \)\(13\!\cdots\!72\)\( T^{5} + \)\(63\!\cdots\!76\)\( T^{6} - \)\(15\!\cdots\!28\)\( T^{7} + \)\(59\!\cdots\!02\)\( T^{8} - \)\(15\!\cdots\!28\)\( p^{8} T^{9} + \)\(63\!\cdots\!76\)\( p^{16} T^{10} - \)\(13\!\cdots\!72\)\( p^{24} T^{11} + 5003239356127079708 p^{32} T^{12} - 81475212513824 p^{40} T^{13} + 2606936824 p^{48} T^{14} - 25696 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
17 \( ( 1 - 13776 T + 26656485624 T^{2} - 437220421682544 T^{3} + \)\(41\!\cdots\!28\)\( T^{4} - \)\(69\!\cdots\!72\)\( T^{5} + \)\(44\!\cdots\!16\)\( T^{6} - \)\(70\!\cdots\!68\)\( T^{7} + \)\(35\!\cdots\!62\)\( T^{8} - \)\(70\!\cdots\!68\)\( p^{8} T^{9} + \)\(44\!\cdots\!16\)\( p^{16} T^{10} - \)\(69\!\cdots\!72\)\( p^{24} T^{11} + \)\(41\!\cdots\!28\)\( p^{32} T^{12} - 437220421682544 p^{40} T^{13} + 26656485624 p^{48} T^{14} - 13776 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
19 \( 1 - 131382634256 T^{2} + \)\(84\!\cdots\!20\)\( T^{4} - \)\(35\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!20\)\( T^{8} - \)\(26\!\cdots\!68\)\( T^{10} + \)\(55\!\cdots\!28\)\( T^{12} - \)\(10\!\cdots\!40\)\( T^{14} + \)\(17\!\cdots\!70\)\( T^{16} - \)\(10\!\cdots\!40\)\( p^{16} T^{18} + \)\(55\!\cdots\!28\)\( p^{32} T^{20} - \)\(26\!\cdots\!68\)\( p^{48} T^{22} + \)\(10\!\cdots\!20\)\( p^{64} T^{24} - \)\(35\!\cdots\!60\)\( p^{80} T^{26} + \)\(84\!\cdots\!20\)\( p^{96} T^{28} - 131382634256 p^{112} T^{30} + p^{128} T^{32} \)
23 \( 1 - 647203509248 T^{2} + \)\(92\!\cdots\!68\)\( p T^{4} - \)\(46\!\cdots\!84\)\( T^{6} + \)\(78\!\cdots\!44\)\( T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(12\!\cdots\!72\)\( T^{12} - \)\(11\!\cdots\!68\)\( T^{14} + \)\(99\!\cdots\!18\)\( T^{16} - \)\(11\!\cdots\!68\)\( p^{16} T^{18} + \)\(12\!\cdots\!72\)\( p^{32} T^{20} - \)\(10\!\cdots\!80\)\( p^{48} T^{22} + \)\(78\!\cdots\!44\)\( p^{64} T^{24} - \)\(46\!\cdots\!84\)\( p^{80} T^{26} + \)\(92\!\cdots\!68\)\( p^{97} T^{28} - 647203509248 p^{112} T^{30} + p^{128} T^{32} \)
29 \( ( 1 - 1382448 T + 2130846542264 T^{2} - 1477340418130986384 T^{3} + \)\(14\!\cdots\!44\)\( T^{4} - \)\(50\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!18\)\( T^{8} + \)\(18\!\cdots\!32\)\( p^{8} T^{9} + \)\(43\!\cdots\!72\)\( p^{16} T^{10} - \)\(50\!\cdots\!80\)\( p^{24} T^{11} + \)\(14\!\cdots\!44\)\( p^{32} T^{12} - 1477340418130986384 p^{40} T^{13} + 2130846542264 p^{48} T^{14} - 1382448 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
31 \( 1 - 9776816431056 T^{2} + \)\(45\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{8} - \)\(48\!\cdots\!68\)\( T^{10} + \)\(65\!\cdots\!28\)\( T^{12} - \)\(73\!\cdots\!40\)\( T^{14} + \)\(67\!\cdots\!70\)\( T^{16} - \)\(73\!\cdots\!40\)\( p^{16} T^{18} + \)\(65\!\cdots\!28\)\( p^{32} T^{20} - \)\(48\!\cdots\!68\)\( p^{48} T^{22} + \)\(29\!\cdots\!20\)\( p^{64} T^{24} - \)\(13\!\cdots\!60\)\( p^{80} T^{26} + \)\(45\!\cdots\!20\)\( p^{96} T^{28} - 9776816431056 p^{112} T^{30} + p^{128} T^{32} \)
37 \( ( 1 - 4504736 T + 27088818847864 T^{2} - 82153294396490747104 T^{3} + \)\(28\!\cdots\!88\)\( T^{4} - \)\(66\!\cdots\!12\)\( T^{5} + \)\(17\!\cdots\!76\)\( T^{6} - \)\(33\!\cdots\!88\)\( T^{7} + \)\(71\!\cdots\!22\)\( T^{8} - \)\(33\!\cdots\!88\)\( p^{8} T^{9} + \)\(17\!\cdots\!76\)\( p^{16} T^{10} - \)\(66\!\cdots\!12\)\( p^{24} T^{11} + \)\(28\!\cdots\!88\)\( p^{32} T^{12} - 82153294396490747104 p^{40} T^{13} + 27088818847864 p^{48} T^{14} - 4504736 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
41 \( ( 1 + 4288224 T + 47153746691320 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(99\!\cdots\!20\)\( T^{4} + \)\(31\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(35\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!70\)\( T^{8} + \)\(35\!\cdots\!60\)\( p^{8} T^{9} + \)\(13\!\cdots\!88\)\( p^{16} T^{10} + \)\(31\!\cdots\!32\)\( p^{24} T^{11} + \)\(99\!\cdots\!20\)\( p^{32} T^{12} + \)\(17\!\cdots\!40\)\( p^{40} T^{13} + 47153746691320 p^{48} T^{14} + 4288224 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
43 \( 1 - 120572816163008 T^{2} + \)\(73\!\cdots\!24\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{6} + \)\(87\!\cdots\!64\)\( T^{8} - \)\(20\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!12\)\( T^{12} - \)\(58\!\cdots\!88\)\( T^{14} + \)\(75\!\cdots\!98\)\( T^{16} - \)\(58\!\cdots\!88\)\( p^{16} T^{18} + \)\(38\!\cdots\!12\)\( p^{32} T^{20} - \)\(20\!\cdots\!80\)\( p^{48} T^{22} + \)\(87\!\cdots\!64\)\( p^{64} T^{24} - \)\(29\!\cdots\!24\)\( p^{80} T^{26} + \)\(73\!\cdots\!24\)\( p^{96} T^{28} - 120572816163008 p^{112} T^{30} + p^{128} T^{32} \)
47 \( 1 - 167993361913088 T^{2} + \)\(14\!\cdots\!04\)\( T^{4} - \)\(80\!\cdots\!44\)\( T^{6} + \)\(34\!\cdots\!24\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(38\!\cdots\!32\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{14} + \)\(26\!\cdots\!38\)\( T^{16} - \)\(10\!\cdots\!48\)\( p^{16} T^{18} + \)\(38\!\cdots\!32\)\( p^{32} T^{20} - \)\(12\!\cdots\!80\)\( p^{48} T^{22} + \)\(34\!\cdots\!24\)\( p^{64} T^{24} - \)\(80\!\cdots\!44\)\( p^{80} T^{26} + \)\(14\!\cdots\!04\)\( p^{96} T^{28} - 167993361913088 p^{112} T^{30} + p^{128} T^{32} \)
53 \( ( 1 - 1226016 T + 172565892414584 T^{2} - \)\(12\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!68\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} - \)\(30\!\cdots\!36\)\( p T^{7} + \)\(11\!\cdots\!62\)\( T^{8} - \)\(30\!\cdots\!36\)\( p^{9} T^{9} + \)\(16\!\cdots\!36\)\( p^{16} T^{10} - \)\(18\!\cdots\!12\)\( p^{24} T^{11} + \)\(17\!\cdots\!68\)\( p^{32} T^{12} - \)\(12\!\cdots\!84\)\( p^{40} T^{13} + 172565892414584 p^{48} T^{14} - 1226016 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
59 \( 1 - 1340721867793936 T^{2} + \)\(87\!\cdots\!20\)\( T^{4} - \)\(36\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!20\)\( T^{8} - \)\(27\!\cdots\!68\)\( T^{10} + \)\(55\!\cdots\!68\)\( T^{12} - \)\(97\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!70\)\( T^{16} - \)\(97\!\cdots\!40\)\( p^{16} T^{18} + \)\(55\!\cdots\!68\)\( p^{32} T^{20} - \)\(27\!\cdots\!68\)\( p^{48} T^{22} + \)\(11\!\cdots\!20\)\( p^{64} T^{24} - \)\(36\!\cdots\!60\)\( p^{80} T^{26} + \)\(87\!\cdots\!20\)\( p^{96} T^{28} - 1340721867793936 p^{112} T^{30} + p^{128} T^{32} \)
61 \( ( 1 - 4185856 T + 834107212763320 T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!20\)\( T^{4} + \)\(15\!\cdots\!32\)\( T^{5} + \)\(52\!\cdots\!28\)\( T^{6} + \)\(76\!\cdots\!60\)\( T^{7} + \)\(81\!\cdots\!70\)\( T^{8} + \)\(76\!\cdots\!60\)\( p^{8} T^{9} + \)\(52\!\cdots\!28\)\( p^{16} T^{10} + \)\(15\!\cdots\!32\)\( p^{24} T^{11} + \)\(28\!\cdots\!20\)\( p^{32} T^{12} - \)\(27\!\cdots\!60\)\( p^{40} T^{13} + 834107212763320 p^{48} T^{14} - 4185856 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
67 \( 1 - 4150786961358528 T^{2} + \)\(79\!\cdots\!44\)\( T^{4} - \)\(93\!\cdots\!04\)\( T^{6} + \)\(73\!\cdots\!04\)\( T^{8} - \)\(40\!\cdots\!80\)\( T^{10} + \)\(15\!\cdots\!92\)\( T^{12} - \)\(45\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!58\)\( T^{16} - \)\(45\!\cdots\!28\)\( p^{16} T^{18} + \)\(15\!\cdots\!92\)\( p^{32} T^{20} - \)\(40\!\cdots\!80\)\( p^{48} T^{22} + \)\(73\!\cdots\!04\)\( p^{64} T^{24} - \)\(93\!\cdots\!04\)\( p^{80} T^{26} + \)\(79\!\cdots\!44\)\( p^{96} T^{28} - 4150786961358528 p^{112} T^{30} + p^{128} T^{32} \)
71 \( 1 - 3525146071156176 T^{2} + \)\(72\!\cdots\!20\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{8} - \)\(13\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{12} - \)\(86\!\cdots\!40\)\( T^{14} + \)\(59\!\cdots\!70\)\( T^{16} - \)\(86\!\cdots\!40\)\( p^{16} T^{18} + \)\(11\!\cdots\!88\)\( p^{32} T^{20} - \)\(13\!\cdots\!68\)\( p^{48} T^{22} + \)\(13\!\cdots\!20\)\( p^{64} T^{24} - \)\(10\!\cdots\!60\)\( p^{80} T^{26} + \)\(72\!\cdots\!20\)\( p^{96} T^{28} - 3525146071156176 p^{112} T^{30} + p^{128} T^{32} \)
73 \( ( 1 - 30953616 T + 4672669211453304 T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(19\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} - \)\(22\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!42\)\( T^{8} - \)\(22\!\cdots\!48\)\( p^{8} T^{9} + \)\(13\!\cdots\!16\)\( p^{16} T^{10} - \)\(19\!\cdots\!72\)\( p^{24} T^{11} + \)\(10\!\cdots\!88\)\( p^{32} T^{12} - \)\(10\!\cdots\!04\)\( p^{40} T^{13} + 4672669211453304 p^{48} T^{14} - 30953616 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
79 \( 1 - 13110568510319376 T^{2} + \)\(87\!\cdots\!20\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!20\)\( T^{8} - \)\(37\!\cdots\!68\)\( T^{10} + \)\(86\!\cdots\!88\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(27\!\cdots\!70\)\( T^{16} - \)\(16\!\cdots\!40\)\( p^{16} T^{18} + \)\(86\!\cdots\!88\)\( p^{32} T^{20} - \)\(37\!\cdots\!68\)\( p^{48} T^{22} + \)\(13\!\cdots\!20\)\( p^{64} T^{24} - \)\(39\!\cdots\!60\)\( p^{80} T^{26} + \)\(87\!\cdots\!20\)\( p^{96} T^{28} - 13110568510319376 p^{112} T^{30} + p^{128} T^{32} \)
83 \( 1 - 17021605643745728 T^{2} + \)\(15\!\cdots\!44\)\( T^{4} - \)\(96\!\cdots\!04\)\( T^{6} + \)\(46\!\cdots\!04\)\( T^{8} - \)\(18\!\cdots\!80\)\( T^{10} + \)\(61\!\cdots\!92\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(42\!\cdots\!58\)\( T^{16} - \)\(17\!\cdots\!28\)\( p^{16} T^{18} + \)\(61\!\cdots\!92\)\( p^{32} T^{20} - \)\(18\!\cdots\!80\)\( p^{48} T^{22} + \)\(46\!\cdots\!04\)\( p^{64} T^{24} - \)\(96\!\cdots\!04\)\( p^{80} T^{26} + \)\(15\!\cdots\!44\)\( p^{96} T^{28} - 17021605643745728 p^{112} T^{30} + p^{128} T^{32} \)
89 \( ( 1 - 53323728 T + 19498736764087544 T^{2} - \)\(52\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!04\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{5} + \)\(92\!\cdots\!92\)\( T^{6} + \)\(10\!\cdots\!72\)\( T^{7} + \)\(39\!\cdots\!58\)\( T^{8} + \)\(10\!\cdots\!72\)\( p^{8} T^{9} + \)\(92\!\cdots\!92\)\( p^{16} T^{10} - \)\(16\!\cdots\!80\)\( p^{24} T^{11} + \)\(16\!\cdots\!04\)\( p^{32} T^{12} - \)\(52\!\cdots\!04\)\( p^{40} T^{13} + 19498736764087544 p^{48} T^{14} - 53323728 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
97 \( ( 1 - 85925616 T + 31055404797673784 T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(54\!\cdots\!28\)\( T^{4} - \)\(39\!\cdots\!72\)\( T^{5} + \)\(65\!\cdots\!96\)\( T^{6} - \)\(42\!\cdots\!68\)\( T^{7} + \)\(59\!\cdots\!62\)\( T^{8} - \)\(42\!\cdots\!68\)\( p^{8} T^{9} + \)\(65\!\cdots\!96\)\( p^{16} T^{10} - \)\(39\!\cdots\!72\)\( p^{24} T^{11} + \)\(54\!\cdots\!28\)\( p^{32} T^{12} - \)\(25\!\cdots\!44\)\( p^{40} T^{13} + 31055404797673784 p^{48} T^{14} - 85925616 p^{56} T^{15} + p^{64} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.28348006125665235449125993286, −3.78346484632197283963486440159, −3.76606897932877496640540239970, −3.54971512733925353034626238994, −3.53993062341067221401711665139, −3.47245825117799841406362489030, −3.44231436407457246983261648793, −3.16383705402529749390716506159, −2.81296862561302362457495402456, −2.63512132988613350818261800083, −2.46627143576614737631621513591, −2.36950947035216275083875431913, −2.36301827240988936462333585362, −2.15695669000228958582541774263, −1.82566659555810617477203785113, −1.74475131993720903041617226492, −1.33712179568876155361826335193, −1.24842739707794167449920408138, −1.19896164369901919403162380118, −1.01668248152558800710877177979, −0.973168131141153608383258824831, −0.827552967984183215476600622154, −0.51309232704318721802565869275, −0.51280250130057676707105705305, −0.37786649778267954848211833995, 0.37786649778267954848211833995, 0.51280250130057676707105705305, 0.51309232704318721802565869275, 0.827552967984183215476600622154, 0.973168131141153608383258824831, 1.01668248152558800710877177979, 1.19896164369901919403162380118, 1.24842739707794167449920408138, 1.33712179568876155361826335193, 1.74475131993720903041617226492, 1.82566659555810617477203785113, 2.15695669000228958582541774263, 2.36301827240988936462333585362, 2.36950947035216275083875431913, 2.46627143576614737631621513591, 2.63512132988613350818261800083, 2.81296862561302362457495402456, 3.16383705402529749390716506159, 3.44231436407457246983261648793, 3.47245825117799841406362489030, 3.53993062341067221401711665139, 3.54971512733925353034626238994, 3.76606897932877496640540239970, 3.78346484632197283963486440159, 4.28348006125665235449125993286

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

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