Properties

Degree $32$
Conductor $6.158\times 10^{28}$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 2·3-s + 15·4-s − 30·5-s − 6·6-s + 56·7-s − 48·8-s − 60·9-s + 90·10-s − 24·11-s + 30·12-s − 68·13-s − 168·14-s − 60·15-s + 202·16-s + 336·17-s + 180·18-s + 352·19-s − 450·20-s + 112·21-s + 72·22-s − 228·23-s − 96·24-s + 828·25-s + 204·26-s − 32·27-s + 840·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.384·3-s + 15/8·4-s − 2.68·5-s − 0.408·6-s + 3.02·7-s − 2.12·8-s − 2.22·9-s + 2.84·10-s − 0.657·11-s + 0.721·12-s − 1.45·13-s − 3.20·14-s − 1.03·15-s + 3.15·16-s + 4.79·17-s + 2.35·18-s + 4.25·19-s − 5.03·20-s + 1.16·21-s + 0.697·22-s − 2.06·23-s − 0.816·24-s + 6.62·25-s + 1.53·26-s − 0.228·27-s + 5.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 7^{16}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.136302\)
\(L(\frac12)\) \(\approx\) \(0.136302\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 2 T + 64 T^{2} - 8 p^{3} T^{3} + 703 p T^{4} - 1462 p^{2} T^{5} + 25 p^{7} T^{6} - 5728 p^{4} T^{7} + 5650 p^{5} T^{8} - 5728 p^{7} T^{9} + 25 p^{13} T^{10} - 1462 p^{11} T^{11} + 703 p^{13} T^{12} - 8 p^{18} T^{13} + 64 p^{18} T^{14} - 2 p^{21} T^{15} + p^{24} T^{16} \)
7 \( ( 1 - p T + p^{2} T^{2} )^{8} \)
good2 \( 1 + 3 T - 3 p T^{2} - 15 T^{3} - 13 T^{4} + 15 T^{5} + 885 T^{6} - 93 p^{2} T^{7} - 1243 p^{3} T^{8} - 1737 p^{2} T^{9} + 5697 T^{10} + 57231 p T^{11} + 70079 p^{2} T^{12} - 12003 p^{7} T^{13} - 161481 p^{4} T^{14} + 114369 p^{6} T^{15} + 354205 p^{6} T^{16} + 114369 p^{9} T^{17} - 161481 p^{10} T^{18} - 12003 p^{16} T^{19} + 70079 p^{14} T^{20} + 57231 p^{16} T^{21} + 5697 p^{18} T^{22} - 1737 p^{23} T^{23} - 1243 p^{27} T^{24} - 93 p^{29} T^{25} + 885 p^{30} T^{26} + 15 p^{33} T^{27} - 13 p^{36} T^{28} - 15 p^{39} T^{29} - 3 p^{43} T^{30} + 3 p^{45} T^{31} + p^{48} T^{32} \)
5 \( 1 + 6 p T + 72 T^{2} - 2088 p T^{3} - 166117 T^{4} + 283968 T^{5} + 7725222 p T^{6} + 93427014 p T^{7} - 196342208 T^{8} - 82657218078 T^{9} - 1137104761158 T^{10} - 2696743553724 T^{11} + 27807034365166 p T^{12} + 450188051017926 p T^{13} + 10174499387616861 T^{14} - 175588804032450084 T^{15} - 3536462896978609646 T^{16} - 175588804032450084 p^{3} T^{17} + 10174499387616861 p^{6} T^{18} + 450188051017926 p^{10} T^{19} + 27807034365166 p^{13} T^{20} - 2696743553724 p^{15} T^{21} - 1137104761158 p^{18} T^{22} - 82657218078 p^{21} T^{23} - 196342208 p^{24} T^{24} + 93427014 p^{28} T^{25} + 7725222 p^{31} T^{26} + 283968 p^{33} T^{27} - 166117 p^{36} T^{28} - 2088 p^{40} T^{29} + 72 p^{42} T^{30} + 6 p^{46} T^{31} + p^{48} T^{32} \)
11 \( 1 + 24 T - 6312 T^{2} - 112236 T^{3} + 20919641 T^{4} + 234941478 T^{5} - 48781328610 T^{6} - 247581606570 T^{7} + 91374825172267 T^{8} + 19514919797334 T^{9} - 148969481247130356 T^{10} + 376103153664918684 T^{11} + 20252741000401342891 p T^{12} - \)\(61\!\cdots\!80\)\( T^{13} - \)\(31\!\cdots\!40\)\( T^{14} + \)\(35\!\cdots\!36\)\( T^{15} + \)\(42\!\cdots\!85\)\( T^{16} + \)\(35\!\cdots\!36\)\( p^{3} T^{17} - \)\(31\!\cdots\!40\)\( p^{6} T^{18} - \)\(61\!\cdots\!80\)\( p^{9} T^{19} + 20252741000401342891 p^{13} T^{20} + 376103153664918684 p^{15} T^{21} - 148969481247130356 p^{18} T^{22} + 19514919797334 p^{21} T^{23} + 91374825172267 p^{24} T^{24} - 247581606570 p^{27} T^{25} - 48781328610 p^{30} T^{26} + 234941478 p^{33} T^{27} + 20919641 p^{36} T^{28} - 112236 p^{39} T^{29} - 6312 p^{42} T^{30} + 24 p^{45} T^{31} + p^{48} T^{32} \)
13 \( 1 + 68 T - 6065 T^{2} - 196604 T^{3} + 34516832 T^{4} - 115057268 T^{5} - 122980759173 T^{6} + 2886205959684 T^{7} + 258268070617101 T^{8} - 13209130095181224 T^{9} - 311290223266507584 T^{10} + 217834710977538120 p^{2} T^{11} - \)\(16\!\cdots\!94\)\( T^{12} - \)\(69\!\cdots\!48\)\( T^{13} + \)\(24\!\cdots\!74\)\( T^{14} + \)\(61\!\cdots\!32\)\( T^{15} - \)\(74\!\cdots\!40\)\( T^{16} + \)\(61\!\cdots\!32\)\( p^{3} T^{17} + \)\(24\!\cdots\!74\)\( p^{6} T^{18} - \)\(69\!\cdots\!48\)\( p^{9} T^{19} - \)\(16\!\cdots\!94\)\( p^{12} T^{20} + 217834710977538120 p^{17} T^{21} - 311290223266507584 p^{18} T^{22} - 13209130095181224 p^{21} T^{23} + 258268070617101 p^{24} T^{24} + 2886205959684 p^{27} T^{25} - 122980759173 p^{30} T^{26} - 115057268 p^{33} T^{27} + 34516832 p^{36} T^{28} - 196604 p^{39} T^{29} - 6065 p^{42} T^{30} + 68 p^{45} T^{31} + p^{48} T^{32} \)
17 \( ( 1 - 168 T + 30394 T^{2} - 3297180 T^{3} + 380502793 T^{4} - 32690504880 T^{5} + 3022684670725 T^{6} - 221800606656432 T^{7} + 17377623608249734 T^{8} - 221800606656432 p^{3} T^{9} + 3022684670725 p^{6} T^{10} - 32690504880 p^{9} T^{11} + 380502793 p^{12} T^{12} - 3297180 p^{15} T^{13} + 30394 p^{18} T^{14} - 168 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
19 \( ( 1 - 176 T + 41892 T^{2} - 4833442 T^{3} + 719223515 T^{4} - 65014349322 T^{5} + 7631904477739 T^{6} - 579970516177592 T^{7} + 59026078922135409 T^{8} - 579970516177592 p^{3} T^{9} + 7631904477739 p^{6} T^{10} - 65014349322 p^{9} T^{11} + 719223515 p^{12} T^{12} - 4833442 p^{15} T^{13} + 41892 p^{18} T^{14} - 176 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
23 \( 1 + 228 T - 25525 T^{2} - 8731368 T^{3} + 389743110 T^{4} + 187567739154 T^{5} - 2655559094585 T^{6} - 2320959795806634 T^{7} + 15687067961480891 T^{8} + 14915354307460983396 T^{9} - \)\(20\!\cdots\!94\)\( T^{10} + \)\(72\!\cdots\!22\)\( T^{11} + \)\(10\!\cdots\!27\)\( T^{12} - \)\(22\!\cdots\!12\)\( T^{13} - \)\(20\!\cdots\!83\)\( T^{14} + \)\(15\!\cdots\!30\)\( T^{15} + \)\(31\!\cdots\!32\)\( T^{16} + \)\(15\!\cdots\!30\)\( p^{3} T^{17} - \)\(20\!\cdots\!83\)\( p^{6} T^{18} - \)\(22\!\cdots\!12\)\( p^{9} T^{19} + \)\(10\!\cdots\!27\)\( p^{12} T^{20} + \)\(72\!\cdots\!22\)\( p^{15} T^{21} - \)\(20\!\cdots\!94\)\( p^{18} T^{22} + 14915354307460983396 p^{21} T^{23} + 15687067961480891 p^{24} T^{24} - 2320959795806634 p^{27} T^{25} - 2655559094585 p^{30} T^{26} + 187567739154 p^{33} T^{27} + 389743110 p^{36} T^{28} - 8731368 p^{39} T^{29} - 25525 p^{42} T^{30} + 228 p^{45} T^{31} + p^{48} T^{32} \)
29 \( 1 + 618 T + 78015 T^{2} - 22761822 T^{3} - 3949828339 T^{4} + 1033935290616 T^{5} + 177028776333588 T^{6} - 27273811167129462 T^{7} - 2315043616387851764 T^{8} + \)\(10\!\cdots\!86\)\( T^{9} - \)\(62\!\cdots\!16\)\( T^{10} - \)\(23\!\cdots\!26\)\( T^{11} + \)\(40\!\cdots\!20\)\( T^{12} + \)\(56\!\cdots\!76\)\( T^{13} - \)\(14\!\cdots\!49\)\( T^{14} - \)\(28\!\cdots\!74\)\( T^{15} + \)\(48\!\cdots\!01\)\( T^{16} - \)\(28\!\cdots\!74\)\( p^{3} T^{17} - \)\(14\!\cdots\!49\)\( p^{6} T^{18} + \)\(56\!\cdots\!76\)\( p^{9} T^{19} + \)\(40\!\cdots\!20\)\( p^{12} T^{20} - \)\(23\!\cdots\!26\)\( p^{15} T^{21} - \)\(62\!\cdots\!16\)\( p^{18} T^{22} + \)\(10\!\cdots\!86\)\( p^{21} T^{23} - 2315043616387851764 p^{24} T^{24} - 27273811167129462 p^{27} T^{25} + 177028776333588 p^{30} T^{26} + 1033935290616 p^{33} T^{27} - 3949828339 p^{36} T^{28} - 22761822 p^{39} T^{29} + 78015 p^{42} T^{30} + 618 p^{45} T^{31} + p^{48} T^{32} \)
31 \( 1 + 72 T - 122345 T^{2} - 411588 p T^{3} + 7008800553 T^{4} + 937649058354 T^{5} - 246288843940504 T^{6} - 39508523240638446 T^{7} + 6189681260162427926 T^{8} + \)\(11\!\cdots\!06\)\( T^{9} - \)\(13\!\cdots\!30\)\( T^{10} - \)\(31\!\cdots\!70\)\( T^{11} + \)\(21\!\cdots\!42\)\( T^{12} + \)\(78\!\cdots\!78\)\( T^{13} + \)\(23\!\cdots\!09\)\( p T^{14} - \)\(99\!\cdots\!06\)\( T^{15} - \)\(13\!\cdots\!55\)\( T^{16} - \)\(99\!\cdots\!06\)\( p^{3} T^{17} + \)\(23\!\cdots\!09\)\( p^{7} T^{18} + \)\(78\!\cdots\!78\)\( p^{9} T^{19} + \)\(21\!\cdots\!42\)\( p^{12} T^{20} - \)\(31\!\cdots\!70\)\( p^{15} T^{21} - \)\(13\!\cdots\!30\)\( p^{18} T^{22} + \)\(11\!\cdots\!06\)\( p^{21} T^{23} + 6189681260162427926 p^{24} T^{24} - 39508523240638446 p^{27} T^{25} - 246288843940504 p^{30} T^{26} + 937649058354 p^{33} T^{27} + 7008800553 p^{36} T^{28} - 411588 p^{40} T^{29} - 122345 p^{42} T^{30} + 72 p^{45} T^{31} + p^{48} T^{32} \)
37 \( ( 1 - 210 T + 273581 T^{2} - 62260500 T^{3} + 37970446276 T^{4} - 8100635347836 T^{5} + 3402478745034716 T^{6} - 624886705874379210 T^{7} + \)\(20\!\cdots\!28\)\( T^{8} - 624886705874379210 p^{3} T^{9} + 3402478745034716 p^{6} T^{10} - 8100635347836 p^{9} T^{11} + 37970446276 p^{12} T^{12} - 62260500 p^{15} T^{13} + 273581 p^{18} T^{14} - 210 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
41 \( 1 + 420 T - 287838 T^{2} - 111388608 T^{3} + 54335362343 T^{4} + 15821993821032 T^{5} - 8270886127177332 T^{6} - 1582199089188412056 T^{7} + \)\(10\!\cdots\!33\)\( T^{8} + \)\(12\!\cdots\!60\)\( T^{9} - \)\(10\!\cdots\!02\)\( T^{10} - \)\(76\!\cdots\!28\)\( T^{11} + \)\(94\!\cdots\!59\)\( T^{12} + \)\(33\!\cdots\!24\)\( T^{13} - \)\(74\!\cdots\!76\)\( T^{14} - \)\(72\!\cdots\!96\)\( T^{15} + \)\(54\!\cdots\!89\)\( T^{16} - \)\(72\!\cdots\!96\)\( p^{3} T^{17} - \)\(74\!\cdots\!76\)\( p^{6} T^{18} + \)\(33\!\cdots\!24\)\( p^{9} T^{19} + \)\(94\!\cdots\!59\)\( p^{12} T^{20} - \)\(76\!\cdots\!28\)\( p^{15} T^{21} - \)\(10\!\cdots\!02\)\( p^{18} T^{22} + \)\(12\!\cdots\!60\)\( p^{21} T^{23} + \)\(10\!\cdots\!33\)\( p^{24} T^{24} - 1582199089188412056 p^{27} T^{25} - 8270886127177332 p^{30} T^{26} + 15821993821032 p^{33} T^{27} + 54335362343 p^{36} T^{28} - 111388608 p^{39} T^{29} - 287838 p^{42} T^{30} + 420 p^{45} T^{31} + p^{48} T^{32} \)
43 \( 1 - 2 T - 174923 T^{2} - 18126802 T^{3} + 6568829129 T^{4} + 3504431725334 T^{5} + 1175920450315236 T^{6} - 266898513634748244 T^{7} - \)\(13\!\cdots\!38\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{9} + \)\(33\!\cdots\!62\)\( T^{10} + \)\(20\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!22\)\( T^{12} - \)\(22\!\cdots\!90\)\( T^{13} - \)\(31\!\cdots\!27\)\( T^{14} + \)\(89\!\cdots\!28\)\( T^{15} + \)\(20\!\cdots\!33\)\( T^{16} + \)\(89\!\cdots\!28\)\( p^{3} T^{17} - \)\(31\!\cdots\!27\)\( p^{6} T^{18} - \)\(22\!\cdots\!90\)\( p^{9} T^{19} + \)\(30\!\cdots\!22\)\( p^{12} T^{20} + \)\(20\!\cdots\!40\)\( p^{15} T^{21} + \)\(33\!\cdots\!62\)\( p^{18} T^{22} - \)\(25\!\cdots\!48\)\( p^{21} T^{23} - \)\(13\!\cdots\!38\)\( p^{24} T^{24} - 266898513634748244 p^{27} T^{25} + 1175920450315236 p^{30} T^{26} + 3504431725334 p^{33} T^{27} + 6568829129 p^{36} T^{28} - 18126802 p^{39} T^{29} - 174923 p^{42} T^{30} - 2 p^{45} T^{31} + p^{48} T^{32} \)
47 \( 1 + 570 T - 504019 T^{2} - 269131554 T^{3} + 178645227255 T^{4} + 75997022312142 T^{5} - 48054772790072906 T^{6} - 15112394704504881348 T^{7} + \)\(10\!\cdots\!88\)\( T^{8} + \)\(22\!\cdots\!32\)\( T^{9} - \)\(18\!\cdots\!92\)\( T^{10} - \)\(26\!\cdots\!90\)\( T^{11} + \)\(28\!\cdots\!88\)\( T^{12} + \)\(22\!\cdots\!72\)\( T^{13} - \)\(37\!\cdots\!29\)\( T^{14} - \)\(88\!\cdots\!70\)\( T^{15} + \)\(41\!\cdots\!29\)\( T^{16} - \)\(88\!\cdots\!70\)\( p^{3} T^{17} - \)\(37\!\cdots\!29\)\( p^{6} T^{18} + \)\(22\!\cdots\!72\)\( p^{9} T^{19} + \)\(28\!\cdots\!88\)\( p^{12} T^{20} - \)\(26\!\cdots\!90\)\( p^{15} T^{21} - \)\(18\!\cdots\!92\)\( p^{18} T^{22} + \)\(22\!\cdots\!32\)\( p^{21} T^{23} + \)\(10\!\cdots\!88\)\( p^{24} T^{24} - 15112394704504881348 p^{27} T^{25} - 48054772790072906 p^{30} T^{26} + 75997022312142 p^{33} T^{27} + 178645227255 p^{36} T^{28} - 269131554 p^{39} T^{29} - 504019 p^{42} T^{30} + 570 p^{45} T^{31} + p^{48} T^{32} \)
53 \( ( 1 - 528 T + 603082 T^{2} - 112356498 T^{3} + 113056847713 T^{4} - 1488977365716 T^{5} + 24102465385714279 T^{6} - 3059793563504889246 T^{7} + \)\(52\!\cdots\!50\)\( T^{8} - 3059793563504889246 p^{3} T^{9} + 24102465385714279 p^{6} T^{10} - 1488977365716 p^{9} T^{11} + 113056847713 p^{12} T^{12} - 112356498 p^{15} T^{13} + 603082 p^{18} T^{14} - 528 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
59 \( 1 - 150 T - 946117 T^{2} + 200189070 T^{3} + 428040629955 T^{4} - 107278598640996 T^{5} - 129587827131610502 T^{6} + 31309587455023028910 T^{7} + \)\(32\!\cdots\!18\)\( T^{8} - \)\(58\!\cdots\!30\)\( T^{9} - \)\(81\!\cdots\!26\)\( T^{10} + \)\(89\!\cdots\!28\)\( T^{11} + \)\(19\!\cdots\!14\)\( T^{12} - \)\(13\!\cdots\!46\)\( T^{13} - \)\(43\!\cdots\!67\)\( T^{14} + \)\(10\!\cdots\!64\)\( T^{15} + \)\(89\!\cdots\!65\)\( T^{16} + \)\(10\!\cdots\!64\)\( p^{3} T^{17} - \)\(43\!\cdots\!67\)\( p^{6} T^{18} - \)\(13\!\cdots\!46\)\( p^{9} T^{19} + \)\(19\!\cdots\!14\)\( p^{12} T^{20} + \)\(89\!\cdots\!28\)\( p^{15} T^{21} - \)\(81\!\cdots\!26\)\( p^{18} T^{22} - \)\(58\!\cdots\!30\)\( p^{21} T^{23} + \)\(32\!\cdots\!18\)\( p^{24} T^{24} + 31309587455023028910 p^{27} T^{25} - 129587827131610502 p^{30} T^{26} - 107278598640996 p^{33} T^{27} + 428040629955 p^{36} T^{28} + 200189070 p^{39} T^{29} - 946117 p^{42} T^{30} - 150 p^{45} T^{31} + p^{48} T^{32} \)
61 \( 1 + 578 T - 1145735 T^{2} - 501446726 T^{3} + 837776641016 T^{4} + 251069047448050 T^{5} - 421088343940353753 T^{6} - 74043629095730306598 T^{7} + \)\(16\!\cdots\!99\)\( T^{8} + \)\(21\!\cdots\!32\)\( p T^{9} - \)\(48\!\cdots\!12\)\( T^{10} - \)\(44\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!09\)\( T^{12} - \)\(29\!\cdots\!26\)\( T^{13} - \)\(26\!\cdots\!71\)\( T^{14} + \)\(45\!\cdots\!78\)\( T^{15} + \)\(57\!\cdots\!72\)\( T^{16} + \)\(45\!\cdots\!78\)\( p^{3} T^{17} - \)\(26\!\cdots\!71\)\( p^{6} T^{18} - \)\(29\!\cdots\!26\)\( p^{9} T^{19} + \)\(11\!\cdots\!09\)\( p^{12} T^{20} - \)\(44\!\cdots\!40\)\( p^{15} T^{21} - \)\(48\!\cdots\!12\)\( p^{18} T^{22} + \)\(21\!\cdots\!32\)\( p^{22} T^{23} + \)\(16\!\cdots\!99\)\( p^{24} T^{24} - 74043629095730306598 p^{27} T^{25} - 421088343940353753 p^{30} T^{26} + 251069047448050 p^{33} T^{27} + 837776641016 p^{36} T^{28} - 501446726 p^{39} T^{29} - 1145735 p^{42} T^{30} + 578 p^{45} T^{31} + p^{48} T^{32} \)
67 \( 1 - 898 T - 791450 T^{2} + 1047042856 T^{3} + 204780930149 T^{4} - 606381046616276 T^{5} + 43407094416136272 T^{6} + \)\(23\!\cdots\!40\)\( T^{7} - \)\(65\!\cdots\!51\)\( T^{8} - \)\(67\!\cdots\!24\)\( T^{9} + \)\(35\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{11} - \)\(13\!\cdots\!05\)\( T^{12} - \)\(17\!\cdots\!58\)\( T^{13} + \)\(38\!\cdots\!12\)\( T^{14} + \)\(11\!\cdots\!04\)\( T^{15} - \)\(11\!\cdots\!23\)\( T^{16} + \)\(11\!\cdots\!04\)\( p^{3} T^{17} + \)\(38\!\cdots\!12\)\( p^{6} T^{18} - \)\(17\!\cdots\!58\)\( p^{9} T^{19} - \)\(13\!\cdots\!05\)\( p^{12} T^{20} + \)\(13\!\cdots\!80\)\( p^{15} T^{21} + \)\(35\!\cdots\!60\)\( p^{18} T^{22} - \)\(67\!\cdots\!24\)\( p^{21} T^{23} - \)\(65\!\cdots\!51\)\( p^{24} T^{24} + \)\(23\!\cdots\!40\)\( p^{27} T^{25} + 43407094416136272 p^{30} T^{26} - 606381046616276 p^{33} T^{27} + 204780930149 p^{36} T^{28} + 1047042856 p^{39} T^{29} - 791450 p^{42} T^{30} - 898 p^{45} T^{31} + p^{48} T^{32} \)
71 \( ( 1 - 882 T + 1565920 T^{2} - 1172203578 T^{3} + 1342300776931 T^{4} - 886834221035832 T^{5} + 765294077320357237 T^{6} - \)\(44\!\cdots\!26\)\( T^{7} + \)\(31\!\cdots\!95\)\( T^{8} - \)\(44\!\cdots\!26\)\( p^{3} T^{9} + 765294077320357237 p^{6} T^{10} - 886834221035832 p^{9} T^{11} + 1342300776931 p^{12} T^{12} - 1172203578 p^{15} T^{13} + 1565920 p^{18} T^{14} - 882 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
73 \( ( 1 - 972 T + 1718351 T^{2} - 1080282636 T^{3} + 1082261782252 T^{4} - 376474331161008 T^{5} + 293371917050729072 T^{6} - 2405960897454683676 T^{7} + \)\(57\!\cdots\!52\)\( T^{8} - 2405960897454683676 p^{3} T^{9} + 293371917050729072 p^{6} T^{10} - 376474331161008 p^{9} T^{11} + 1082261782252 p^{12} T^{12} - 1080282636 p^{15} T^{13} + 1718351 p^{18} T^{14} - 972 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
79 \( 1 - 2 p T - 1631978 T^{2} + 326581604 T^{3} + 1003734307967 T^{4} - 276658431385546 T^{5} - 294010705461071412 T^{6} + \)\(16\!\cdots\!94\)\( T^{7} + \)\(10\!\cdots\!84\)\( T^{8} - \)\(77\!\cdots\!36\)\( T^{9} - \)\(89\!\cdots\!86\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(53\!\cdots\!98\)\( T^{12} + \)\(92\!\cdots\!86\)\( T^{13} - \)\(20\!\cdots\!95\)\( T^{14} - \)\(39\!\cdots\!96\)\( T^{15} + \)\(81\!\cdots\!30\)\( T^{16} - \)\(39\!\cdots\!96\)\( p^{3} T^{17} - \)\(20\!\cdots\!95\)\( p^{6} T^{18} + \)\(92\!\cdots\!86\)\( p^{9} T^{19} + \)\(53\!\cdots\!98\)\( p^{12} T^{20} + \)\(15\!\cdots\!56\)\( p^{15} T^{21} - \)\(89\!\cdots\!86\)\( p^{18} T^{22} - \)\(77\!\cdots\!36\)\( p^{21} T^{23} + \)\(10\!\cdots\!84\)\( p^{24} T^{24} + \)\(16\!\cdots\!94\)\( p^{27} T^{25} - 294010705461071412 p^{30} T^{26} - 276658431385546 p^{33} T^{27} + 1003734307967 p^{36} T^{28} + 326581604 p^{39} T^{29} - 1631978 p^{42} T^{30} - 2 p^{46} T^{31} + p^{48} T^{32} \)
83 \( 1 + 2958 T + 2046194 T^{2} - 1191819780 T^{3} - 156888285003 T^{4} + 2954536317243036 T^{5} + 1354865827302829804 T^{6} - \)\(96\!\cdots\!52\)\( T^{7} + \)\(25\!\cdots\!17\)\( T^{8} + \)\(90\!\cdots\!24\)\( T^{9} - \)\(57\!\cdots\!04\)\( T^{10} - \)\(67\!\cdots\!52\)\( T^{11} + \)\(24\!\cdots\!15\)\( T^{12} - \)\(26\!\cdots\!58\)\( T^{13} - \)\(13\!\cdots\!52\)\( T^{14} + \)\(69\!\cdots\!52\)\( T^{15} + \)\(56\!\cdots\!77\)\( T^{16} + \)\(69\!\cdots\!52\)\( p^{3} T^{17} - \)\(13\!\cdots\!52\)\( p^{6} T^{18} - \)\(26\!\cdots\!58\)\( p^{9} T^{19} + \)\(24\!\cdots\!15\)\( p^{12} T^{20} - \)\(67\!\cdots\!52\)\( p^{15} T^{21} - \)\(57\!\cdots\!04\)\( p^{18} T^{22} + \)\(90\!\cdots\!24\)\( p^{21} T^{23} + \)\(25\!\cdots\!17\)\( p^{24} T^{24} - \)\(96\!\cdots\!52\)\( p^{27} T^{25} + 1354865827302829804 p^{30} T^{26} + 2954536317243036 p^{33} T^{27} - 156888285003 p^{36} T^{28} - 1191819780 p^{39} T^{29} + 2046194 p^{42} T^{30} + 2958 p^{45} T^{31} + p^{48} T^{32} \)
89 \( ( 1 - 4380 T + 10823694 T^{2} - 19914754776 T^{3} + 29899723956313 T^{4} - 38260156523638452 T^{5} + 42933634978495882653 T^{6} - \)\(42\!\cdots\!56\)\( T^{7} + \)\(37\!\cdots\!62\)\( T^{8} - \)\(42\!\cdots\!56\)\( p^{3} T^{9} + 42933634978495882653 p^{6} T^{10} - 38260156523638452 p^{9} T^{11} + 29899723956313 p^{12} T^{12} - 19914754776 p^{15} T^{13} + 10823694 p^{18} T^{14} - 4380 p^{21} T^{15} + p^{24} T^{16} )^{2} \)
97 \( 1 - 60 T - 3109535 T^{2} + 2828204676 T^{3} + 4865749520445 T^{4} - 7975237084169220 T^{5} - 624844777429079932 T^{6} + \)\(11\!\cdots\!12\)\( T^{7} - \)\(80\!\cdots\!40\)\( T^{8} - \)\(65\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!28\)\( T^{10} - \)\(36\!\cdots\!20\)\( T^{11} - \)\(86\!\cdots\!92\)\( T^{12} + \)\(97\!\cdots\!76\)\( T^{13} - \)\(16\!\cdots\!31\)\( T^{14} - \)\(45\!\cdots\!64\)\( T^{15} + \)\(67\!\cdots\!49\)\( T^{16} - \)\(45\!\cdots\!64\)\( p^{3} T^{17} - \)\(16\!\cdots\!31\)\( p^{6} T^{18} + \)\(97\!\cdots\!76\)\( p^{9} T^{19} - \)\(86\!\cdots\!92\)\( p^{12} T^{20} - \)\(36\!\cdots\!20\)\( p^{15} T^{21} + \)\(13\!\cdots\!28\)\( p^{18} T^{22} - \)\(65\!\cdots\!40\)\( p^{21} T^{23} - \)\(80\!\cdots\!40\)\( p^{24} T^{24} + \)\(11\!\cdots\!12\)\( p^{27} T^{25} - 624844777429079932 p^{30} T^{26} - 7975237084169220 p^{33} T^{27} + 4865749520445 p^{36} T^{28} + 2828204676 p^{39} T^{29} - 3109535 p^{42} T^{30} - 60 p^{45} T^{31} + p^{48} T^{32} \)
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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

−4.15144209598724826996625175124, −3.77458224081294579482075298037, −3.71430661312872125517713898861, −3.60995909498732119252967365325, −3.45121627763063097762332086733, −3.45090934989475781108865918861, −3.43612368671424180778522189786, −3.36674174827280871276542831336, −3.23122089043815030542848849651, −3.18971955606463887201153533825, −3.00260165427286123950953129186, −2.67535155878948797692090943778, −2.51054392916014174353828992722, −2.40098738804147897123503400960, −2.35594366059793525635554377309, −2.20914620181221725350415928445, −2.00542651801110699578233678277, −1.58228085817417117343241623447, −1.57228710194750973228350208005, −1.40194691810868791003134941779, −0.977155801124939165433842024963, −0.855898452752791071067842181422, −0.77697921606178826144602194153, −0.72934243061518344713663861411, −0.03421748582433319819363761749, 0.03421748582433319819363761749, 0.72934243061518344713663861411, 0.77697921606178826144602194153, 0.855898452752791071067842181422, 0.977155801124939165433842024963, 1.40194691810868791003134941779, 1.57228710194750973228350208005, 1.58228085817417117343241623447, 2.00542651801110699578233678277, 2.20914620181221725350415928445, 2.35594366059793525635554377309, 2.40098738804147897123503400960, 2.51054392916014174353828992722, 2.67535155878948797692090943778, 3.00260165427286123950953129186, 3.18971955606463887201153533825, 3.23122089043815030542848849651, 3.36674174827280871276542831336, 3.43612368671424180778522189786, 3.45090934989475781108865918861, 3.45121627763063097762332086733, 3.60995909498732119252967365325, 3.71430661312872125517713898861, 3.77458224081294579482075298037, 4.15144209598724826996625175124

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

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