Properties

Label 28-1875e14-1.1-c3e14-0-3
Degree $28$
Conductor $6.638\times 10^{45}$
Sign $1$
Analytic cond. $4.11284\times 10^{28}$
Root an. cond. $10.5180$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $14$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 42·3-s − 31·4-s − 27·7-s + 7·8-s + 945·9-s − 33·11-s − 1.30e3·12-s − 188·13-s + 401·16-s − 146·17-s − 184·19-s − 1.13e3·21-s − 164·23-s + 294·24-s + 1.51e4·27-s + 837·28-s + 252·29-s − 889·31-s − 335·32-s − 1.38e3·33-s − 2.92e4·36-s − 642·37-s − 7.89e3·39-s − 164·41-s − 696·43-s + 1.02e3·44-s + 92·47-s + ⋯
L(s)  = 1  + 8.08·3-s − 3.87·4-s − 1.45·7-s + 0.309·8-s + 35·9-s − 0.904·11-s − 31.3·12-s − 4.01·13-s + 6.26·16-s − 2.08·17-s − 2.22·19-s − 11.7·21-s − 1.48·23-s + 2.50·24-s + 107.·27-s + 5.64·28-s + 1.61·29-s − 5.15·31-s − 1.85·32-s − 7.31·33-s − 135.·36-s − 2.85·37-s − 32.4·39-s − 0.624·41-s − 2.46·43-s + 3.50·44-s + 0.285·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{56}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{56}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(3^{14} \cdot 5^{56}\)
Sign: $1$
Analytic conductor: \(4.11284\times 10^{28}\)
Root analytic conductor: \(10.5180\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(14\)
Selberg data: \((28,\ 3^{14} \cdot 5^{56} ,\ ( \ : [3/2]^{14} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p T )^{14} \)
5 \( 1 \)
good2 \( 1 + 31 T^{2} - 7 T^{3} + 35 p^{4} T^{4} - 99 T^{5} + 4107 p T^{6} - 403 T^{7} + 102027 T^{8} - 1685 T^{9} + 1090711 T^{10} - 10871 p T^{11} + 1298021 p^{3} T^{12} + 737 p^{7} T^{13} + 5525365 p^{4} T^{14} + 737 p^{10} T^{15} + 1298021 p^{9} T^{16} - 10871 p^{10} T^{17} + 1090711 p^{12} T^{18} - 1685 p^{15} T^{19} + 102027 p^{18} T^{20} - 403 p^{21} T^{21} + 4107 p^{25} T^{22} - 99 p^{27} T^{23} + 35 p^{34} T^{24} - 7 p^{33} T^{25} + 31 p^{36} T^{26} + p^{42} T^{28} \)
7 \( 1 + 27 T + 2725 T^{2} + 71247 T^{3} + 3778737 T^{4} + 92712940 T^{5} + 3525750533 T^{6} + 79961548059 T^{7} + 2463712450179 T^{8} + 51433568663703 T^{9} + 3971342571190 p^{3} T^{10} + 3738557819688691 p T^{11} + 1793182984025057 p^{3} T^{12} + 10860188118458403801 T^{13} + \)\(23\!\cdots\!12\)\( T^{14} + 10860188118458403801 p^{3} T^{15} + 1793182984025057 p^{9} T^{16} + 3738557819688691 p^{10} T^{17} + 3971342571190 p^{15} T^{18} + 51433568663703 p^{15} T^{19} + 2463712450179 p^{18} T^{20} + 79961548059 p^{21} T^{21} + 3525750533 p^{24} T^{22} + 92712940 p^{27} T^{23} + 3778737 p^{30} T^{24} + 71247 p^{33} T^{25} + 2725 p^{36} T^{26} + 27 p^{39} T^{27} + p^{42} T^{28} \)
11 \( 1 + 3 p T + 8672 T^{2} + 283194 T^{3} + 39712880 T^{4} + 1302996351 T^{5} + 128847611378 T^{6} + 4173260457477 T^{7} + 326038266870629 T^{8} + 10191899938382955 T^{9} + 671735607979006957 T^{10} + 19912832190934114296 T^{11} + \)\(11\!\cdots\!14\)\( T^{12} + \)\(31\!\cdots\!88\)\( T^{13} + \)\(16\!\cdots\!30\)\( T^{14} + \)\(31\!\cdots\!88\)\( p^{3} T^{15} + \)\(11\!\cdots\!14\)\( p^{6} T^{16} + 19912832190934114296 p^{9} T^{17} + 671735607979006957 p^{12} T^{18} + 10191899938382955 p^{15} T^{19} + 326038266870629 p^{18} T^{20} + 4173260457477 p^{21} T^{21} + 128847611378 p^{24} T^{22} + 1302996351 p^{27} T^{23} + 39712880 p^{30} T^{24} + 283194 p^{33} T^{25} + 8672 p^{36} T^{26} + 3 p^{40} T^{27} + p^{42} T^{28} \)
13 \( 1 + 188 T + 25946 T^{2} + 2544198 T^{3} + 210316949 T^{4} + 14622184948 T^{5} + 908988191895 T^{6} + 51124059157566 T^{7} + 2715782301163382 T^{8} + 139828208151410108 T^{9} + 7211117776811173063 T^{10} + \)\(37\!\cdots\!36\)\( T^{11} + \)\(19\!\cdots\!09\)\( T^{12} + \)\(97\!\cdots\!60\)\( T^{13} + \)\(46\!\cdots\!94\)\( T^{14} + \)\(97\!\cdots\!60\)\( p^{3} T^{15} + \)\(19\!\cdots\!09\)\( p^{6} T^{16} + \)\(37\!\cdots\!36\)\( p^{9} T^{17} + 7211117776811173063 p^{12} T^{18} + 139828208151410108 p^{15} T^{19} + 2715782301163382 p^{18} T^{20} + 51124059157566 p^{21} T^{21} + 908988191895 p^{24} T^{22} + 14622184948 p^{27} T^{23} + 210316949 p^{30} T^{24} + 2544198 p^{33} T^{25} + 25946 p^{36} T^{26} + 188 p^{39} T^{27} + p^{42} T^{28} \)
17 \( 1 + 146 T + 49763 T^{2} + 6020000 T^{3} + 1160642866 T^{4} + 120297444738 T^{5} + 17164831878995 T^{6} + 1563769075868606 T^{7} + 183161188175350836 T^{8} + 14939778132018520708 T^{9} + \)\(15\!\cdots\!97\)\( T^{10} + \)\(11\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!42\)\( T^{12} + \)\(67\!\cdots\!68\)\( T^{13} + \)\(54\!\cdots\!52\)\( T^{14} + \)\(67\!\cdots\!68\)\( p^{3} T^{15} + \)\(10\!\cdots\!42\)\( p^{6} T^{16} + \)\(11\!\cdots\!48\)\( p^{9} T^{17} + \)\(15\!\cdots\!97\)\( p^{12} T^{18} + 14939778132018520708 p^{15} T^{19} + 183161188175350836 p^{18} T^{20} + 1563769075868606 p^{21} T^{21} + 17164831878995 p^{24} T^{22} + 120297444738 p^{27} T^{23} + 1160642866 p^{30} T^{24} + 6020000 p^{33} T^{25} + 49763 p^{36} T^{26} + 146 p^{39} T^{27} + p^{42} T^{28} \)
19 \( 1 + 184 T + 57671 T^{2} + 7768588 T^{3} + 78237824 p T^{4} + 164884057906 T^{5} + 1292136656313 p T^{6} + 2356900502376800 T^{7} + 298277733754628153 T^{8} + 25467185873673376764 T^{9} + \)\(28\!\cdots\!71\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{11} + \)\(23\!\cdots\!62\)\( T^{12} + \)\(17\!\cdots\!78\)\( T^{13} + \)\(16\!\cdots\!58\)\( T^{14} + \)\(17\!\cdots\!78\)\( p^{3} T^{15} + \)\(23\!\cdots\!62\)\( p^{6} T^{16} + \)\(22\!\cdots\!16\)\( p^{9} T^{17} + \)\(28\!\cdots\!71\)\( p^{12} T^{18} + 25467185873673376764 p^{15} T^{19} + 298277733754628153 p^{18} T^{20} + 2356900502376800 p^{21} T^{21} + 1292136656313 p^{25} T^{22} + 164884057906 p^{27} T^{23} + 78237824 p^{31} T^{24} + 7768588 p^{33} T^{25} + 57671 p^{36} T^{26} + 184 p^{39} T^{27} + p^{42} T^{28} \)
23 \( 1 + 164 T + 4525 p T^{2} + 14230126 T^{3} + 229824449 p T^{4} + 636085994080 T^{5} + 178138385270772 T^{6} + 19349653137178632 T^{7} + 4479188365247547944 T^{8} + 19304704352885913682 p T^{9} + \)\(88\!\cdots\!15\)\( T^{10} + \)\(80\!\cdots\!16\)\( T^{11} + \)\(14\!\cdots\!76\)\( T^{12} + \)\(11\!\cdots\!32\)\( T^{13} + \)\(19\!\cdots\!08\)\( T^{14} + \)\(11\!\cdots\!32\)\( p^{3} T^{15} + \)\(14\!\cdots\!76\)\( p^{6} T^{16} + \)\(80\!\cdots\!16\)\( p^{9} T^{17} + \)\(88\!\cdots\!15\)\( p^{12} T^{18} + 19304704352885913682 p^{16} T^{19} + 4479188365247547944 p^{18} T^{20} + 19349653137178632 p^{21} T^{21} + 178138385270772 p^{24} T^{22} + 636085994080 p^{27} T^{23} + 229824449 p^{31} T^{24} + 14230126 p^{33} T^{25} + 4525 p^{37} T^{26} + 164 p^{39} T^{27} + p^{42} T^{28} \)
29 \( 1 - 252 T + 9179 p T^{2} - 60250594 T^{3} + 33861564586 T^{4} - 6879637029558 T^{5} + 2732911476565325 T^{6} - 499244534286052480 T^{7} + 5411299999218938726 p T^{8} - \)\(25\!\cdots\!02\)\( T^{9} + \)\(68\!\cdots\!75\)\( T^{10} - \)\(10\!\cdots\!28\)\( T^{11} + \)\(23\!\cdots\!04\)\( T^{12} - \)\(31\!\cdots\!34\)\( T^{13} + \)\(63\!\cdots\!48\)\( T^{14} - \)\(31\!\cdots\!34\)\( p^{3} T^{15} + \)\(23\!\cdots\!04\)\( p^{6} T^{16} - \)\(10\!\cdots\!28\)\( p^{9} T^{17} + \)\(68\!\cdots\!75\)\( p^{12} T^{18} - \)\(25\!\cdots\!02\)\( p^{15} T^{19} + 5411299999218938726 p^{19} T^{20} - 499244534286052480 p^{21} T^{21} + 2732911476565325 p^{24} T^{22} - 6879637029558 p^{27} T^{23} + 33861564586 p^{30} T^{24} - 60250594 p^{33} T^{25} + 9179 p^{37} T^{26} - 252 p^{39} T^{27} + p^{42} T^{28} \)
31 \( 1 + 889 T + 538276 T^{2} + 241730144 T^{3} + 92486349950 T^{4} + 30757442587627 T^{5} + 9254986299299708 T^{6} + 2536866498404369487 T^{7} + \)\(64\!\cdots\!23\)\( T^{8} + \)\(15\!\cdots\!85\)\( T^{9} + \)\(33\!\cdots\!09\)\( T^{10} + \)\(22\!\cdots\!76\)\( p T^{11} + \)\(13\!\cdots\!74\)\( T^{12} + \)\(26\!\cdots\!26\)\( T^{13} + \)\(46\!\cdots\!30\)\( T^{14} + \)\(26\!\cdots\!26\)\( p^{3} T^{15} + \)\(13\!\cdots\!74\)\( p^{6} T^{16} + \)\(22\!\cdots\!76\)\( p^{10} T^{17} + \)\(33\!\cdots\!09\)\( p^{12} T^{18} + \)\(15\!\cdots\!85\)\( p^{15} T^{19} + \)\(64\!\cdots\!23\)\( p^{18} T^{20} + 2536866498404369487 p^{21} T^{21} + 9254986299299708 p^{24} T^{22} + 30757442587627 p^{27} T^{23} + 92486349950 p^{30} T^{24} + 241730144 p^{33} T^{25} + 538276 p^{36} T^{26} + 889 p^{39} T^{27} + p^{42} T^{28} \)
37 \( 1 + 642 T + 358005 T^{2} + 117291442 T^{3} + 39961349042 T^{4} + 283057142730 p T^{5} + 3456561792031563 T^{6} + 931271982550302384 T^{7} + \)\(28\!\cdots\!14\)\( T^{8} + \)\(68\!\cdots\!38\)\( T^{9} + \)\(18\!\cdots\!75\)\( T^{10} + \)\(41\!\cdots\!22\)\( T^{11} + \)\(10\!\cdots\!76\)\( T^{12} + \)\(23\!\cdots\!06\)\( T^{13} + \)\(60\!\cdots\!52\)\( T^{14} + \)\(23\!\cdots\!06\)\( p^{3} T^{15} + \)\(10\!\cdots\!76\)\( p^{6} T^{16} + \)\(41\!\cdots\!22\)\( p^{9} T^{17} + \)\(18\!\cdots\!75\)\( p^{12} T^{18} + \)\(68\!\cdots\!38\)\( p^{15} T^{19} + \)\(28\!\cdots\!14\)\( p^{18} T^{20} + 931271982550302384 p^{21} T^{21} + 3456561792031563 p^{24} T^{22} + 283057142730 p^{28} T^{23} + 39961349042 p^{30} T^{24} + 117291442 p^{33} T^{25} + 358005 p^{36} T^{26} + 642 p^{39} T^{27} + p^{42} T^{28} \)
41 \( 1 + 4 p T + 562021 T^{2} + 111522544 T^{3} + 159295177490 T^{4} + 33486019573242 T^{5} + 30286609525646258 T^{6} + 6251984895096299942 T^{7} + \)\(42\!\cdots\!43\)\( T^{8} + \)\(83\!\cdots\!40\)\( T^{9} + \)\(47\!\cdots\!64\)\( T^{10} + \)\(87\!\cdots\!36\)\( T^{11} + \)\(43\!\cdots\!99\)\( T^{12} + \)\(72\!\cdots\!46\)\( T^{13} + \)\(32\!\cdots\!40\)\( T^{14} + \)\(72\!\cdots\!46\)\( p^{3} T^{15} + \)\(43\!\cdots\!99\)\( p^{6} T^{16} + \)\(87\!\cdots\!36\)\( p^{9} T^{17} + \)\(47\!\cdots\!64\)\( p^{12} T^{18} + \)\(83\!\cdots\!40\)\( p^{15} T^{19} + \)\(42\!\cdots\!43\)\( p^{18} T^{20} + 6251984895096299942 p^{21} T^{21} + 30286609525646258 p^{24} T^{22} + 33486019573242 p^{27} T^{23} + 159295177490 p^{30} T^{24} + 111522544 p^{33} T^{25} + 562021 p^{36} T^{26} + 4 p^{40} T^{27} + p^{42} T^{28} \)
43 \( 1 + 696 T + 893688 T^{2} + 506713902 T^{3} + 365367808208 T^{4} + 174607176860894 T^{5} + 92367241824009482 T^{6} + 38177743112109680738 T^{7} + \)\(16\!\cdots\!01\)\( T^{8} + \)\(59\!\cdots\!80\)\( T^{9} + \)\(22\!\cdots\!70\)\( T^{10} + \)\(72\!\cdots\!10\)\( T^{11} + \)\(23\!\cdots\!50\)\( T^{12} + \)\(70\!\cdots\!80\)\( T^{13} + \)\(20\!\cdots\!20\)\( T^{14} + \)\(70\!\cdots\!80\)\( p^{3} T^{15} + \)\(23\!\cdots\!50\)\( p^{6} T^{16} + \)\(72\!\cdots\!10\)\( p^{9} T^{17} + \)\(22\!\cdots\!70\)\( p^{12} T^{18} + \)\(59\!\cdots\!80\)\( p^{15} T^{19} + \)\(16\!\cdots\!01\)\( p^{18} T^{20} + 38177743112109680738 p^{21} T^{21} + 92367241824009482 p^{24} T^{22} + 174607176860894 p^{27} T^{23} + 365367808208 p^{30} T^{24} + 506713902 p^{33} T^{25} + 893688 p^{36} T^{26} + 696 p^{39} T^{27} + p^{42} T^{28} \)
47 \( 1 - 92 T + 843512 T^{2} - 87434456 T^{3} + 360865250413 T^{4} - 40200419978968 T^{5} + 103582686359307753 T^{6} - 11847937785932970204 T^{7} + \)\(22\!\cdots\!86\)\( T^{8} - \)\(25\!\cdots\!70\)\( T^{9} + \)\(37\!\cdots\!70\)\( T^{10} - \)\(40\!\cdots\!10\)\( T^{11} + \)\(52\!\cdots\!40\)\( T^{12} - \)\(52\!\cdots\!00\)\( T^{13} + \)\(59\!\cdots\!50\)\( T^{14} - \)\(52\!\cdots\!00\)\( p^{3} T^{15} + \)\(52\!\cdots\!40\)\( p^{6} T^{16} - \)\(40\!\cdots\!10\)\( p^{9} T^{17} + \)\(37\!\cdots\!70\)\( p^{12} T^{18} - \)\(25\!\cdots\!70\)\( p^{15} T^{19} + \)\(22\!\cdots\!86\)\( p^{18} T^{20} - 11847937785932970204 p^{21} T^{21} + 103582686359307753 p^{24} T^{22} - 40200419978968 p^{27} T^{23} + 360865250413 p^{30} T^{24} - 87434456 p^{33} T^{25} + 843512 p^{36} T^{26} - 92 p^{39} T^{27} + p^{42} T^{28} \)
53 \( 1 + 949 T + 1569160 T^{2} + 998341966 T^{3} + 957679438527 T^{4} + 439929739954990 T^{5} + 324278540240954332 T^{6} + \)\(11\!\cdots\!77\)\( T^{7} + \)\(14\!\cdots\!38\)\( p T^{8} + \)\(20\!\cdots\!51\)\( T^{9} + \)\(14\!\cdots\!90\)\( T^{10} + \)\(36\!\cdots\!36\)\( T^{11} + \)\(28\!\cdots\!01\)\( T^{12} + \)\(66\!\cdots\!62\)\( T^{13} + \)\(46\!\cdots\!63\)\( T^{14} + \)\(66\!\cdots\!62\)\( p^{3} T^{15} + \)\(28\!\cdots\!01\)\( p^{6} T^{16} + \)\(36\!\cdots\!36\)\( p^{9} T^{17} + \)\(14\!\cdots\!90\)\( p^{12} T^{18} + \)\(20\!\cdots\!51\)\( p^{15} T^{19} + \)\(14\!\cdots\!38\)\( p^{19} T^{20} + \)\(11\!\cdots\!77\)\( p^{21} T^{21} + 324278540240954332 p^{24} T^{22} + 439929739954990 p^{27} T^{23} + 957679438527 p^{30} T^{24} + 998341966 p^{33} T^{25} + 1569160 p^{36} T^{26} + 949 p^{39} T^{27} + p^{42} T^{28} \)
59 \( 1 + 81 T + 1303786 T^{2} + 92580452 T^{3} + 16160316939 p T^{4} + 57365285875134 T^{5} + 488296045384852962 T^{6} + 24720172464815019465 T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(83\!\cdots\!46\)\( T^{9} + \)\(61\!\cdots\!81\)\( T^{10} + \)\(23\!\cdots\!09\)\( T^{11} + \)\(16\!\cdots\!37\)\( T^{12} + \)\(55\!\cdots\!77\)\( T^{13} + \)\(36\!\cdots\!78\)\( T^{14} + \)\(55\!\cdots\!77\)\( p^{3} T^{15} + \)\(16\!\cdots\!37\)\( p^{6} T^{16} + \)\(23\!\cdots\!09\)\( p^{9} T^{17} + \)\(61\!\cdots\!81\)\( p^{12} T^{18} + \)\(83\!\cdots\!46\)\( p^{15} T^{19} + \)\(19\!\cdots\!13\)\( p^{18} T^{20} + 24720172464815019465 p^{21} T^{21} + 488296045384852962 p^{24} T^{22} + 57365285875134 p^{27} T^{23} + 16160316939 p^{31} T^{24} + 92580452 p^{33} T^{25} + 1303786 p^{36} T^{26} + 81 p^{39} T^{27} + p^{42} T^{28} \)
61 \( 1 + 496 T + 1611629 T^{2} + 717373524 T^{3} + 1254758311590 T^{4} + 500912579275194 T^{5} + 631061393751929954 T^{6} + \)\(22\!\cdots\!86\)\( T^{7} + \)\(23\!\cdots\!91\)\( T^{8} + \)\(76\!\cdots\!00\)\( T^{9} + \)\(69\!\cdots\!28\)\( T^{10} + \)\(21\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!07\)\( T^{12} + \)\(84\!\cdots\!82\)\( p T^{13} + \)\(42\!\cdots\!20\)\( T^{14} + \)\(84\!\cdots\!82\)\( p^{4} T^{15} + \)\(17\!\cdots\!07\)\( p^{6} T^{16} + \)\(21\!\cdots\!08\)\( p^{9} T^{17} + \)\(69\!\cdots\!28\)\( p^{12} T^{18} + \)\(76\!\cdots\!00\)\( p^{15} T^{19} + \)\(23\!\cdots\!91\)\( p^{18} T^{20} + \)\(22\!\cdots\!86\)\( p^{21} T^{21} + 631061393751929954 p^{24} T^{22} + 500912579275194 p^{27} T^{23} + 1254758311590 p^{30} T^{24} + 717373524 p^{33} T^{25} + 1611629 p^{36} T^{26} + 496 p^{39} T^{27} + p^{42} T^{28} \)
67 \( 1 + 1926 T + 4003828 T^{2} + 5114059640 T^{3} + 6524797783036 T^{4} + 6509570001454308 T^{5} + 6403688756244027970 T^{6} + \)\(53\!\cdots\!36\)\( T^{7} + \)\(44\!\cdots\!81\)\( T^{8} + \)\(32\!\cdots\!68\)\( T^{9} + \)\(23\!\cdots\!82\)\( T^{10} + \)\(14\!\cdots\!38\)\( T^{11} + \)\(95\!\cdots\!02\)\( T^{12} + \)\(55\!\cdots\!48\)\( T^{13} + \)\(31\!\cdots\!92\)\( T^{14} + \)\(55\!\cdots\!48\)\( p^{3} T^{15} + \)\(95\!\cdots\!02\)\( p^{6} T^{16} + \)\(14\!\cdots\!38\)\( p^{9} T^{17} + \)\(23\!\cdots\!82\)\( p^{12} T^{18} + \)\(32\!\cdots\!68\)\( p^{15} T^{19} + \)\(44\!\cdots\!81\)\( p^{18} T^{20} + \)\(53\!\cdots\!36\)\( p^{21} T^{21} + 6403688756244027970 p^{24} T^{22} + 6509570001454308 p^{27} T^{23} + 6524797783036 p^{30} T^{24} + 5114059640 p^{33} T^{25} + 4003828 p^{36} T^{26} + 1926 p^{39} T^{27} + p^{42} T^{28} \)
71 \( 1 - 2498 T + 81678 p T^{2} - 8757063456 T^{3} + 12273968496945 T^{4} - 13842707296747820 T^{5} + 14760954603678774965 T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!90\)\( T^{8} - \)\(96\!\cdots\!80\)\( T^{9} + \)\(74\!\cdots\!20\)\( T^{10} - \)\(52\!\cdots\!30\)\( T^{11} + \)\(36\!\cdots\!80\)\( T^{12} - \)\(23\!\cdots\!00\)\( T^{13} + \)\(14\!\cdots\!70\)\( T^{14} - \)\(23\!\cdots\!00\)\( p^{3} T^{15} + \)\(36\!\cdots\!80\)\( p^{6} T^{16} - \)\(52\!\cdots\!30\)\( p^{9} T^{17} + \)\(74\!\cdots\!20\)\( p^{12} T^{18} - \)\(96\!\cdots\!80\)\( p^{15} T^{19} + \)\(12\!\cdots\!90\)\( p^{18} T^{20} - \)\(13\!\cdots\!40\)\( p^{21} T^{21} + 14760954603678774965 p^{24} T^{22} - 13842707296747820 p^{27} T^{23} + 12273968496945 p^{30} T^{24} - 8757063456 p^{33} T^{25} + 81678 p^{37} T^{26} - 2498 p^{39} T^{27} + p^{42} T^{28} \)
73 \( 1 - 1026 T + 3107635 T^{2} - 2797052904 T^{3} + 4769157286117 T^{4} - 3842038823167800 T^{5} + 4895660342061304707 T^{6} - \)\(35\!\cdots\!18\)\( T^{7} + \)\(37\!\cdots\!69\)\( T^{8} - \)\(25\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!30\)\( T^{10} - \)\(14\!\cdots\!44\)\( T^{11} + \)\(11\!\cdots\!46\)\( T^{12} - \)\(67\!\cdots\!88\)\( T^{13} + \)\(50\!\cdots\!78\)\( T^{14} - \)\(67\!\cdots\!88\)\( p^{3} T^{15} + \)\(11\!\cdots\!46\)\( p^{6} T^{16} - \)\(14\!\cdots\!44\)\( p^{9} T^{17} + \)\(23\!\cdots\!30\)\( p^{12} T^{18} - \)\(25\!\cdots\!24\)\( p^{15} T^{19} + \)\(37\!\cdots\!69\)\( p^{18} T^{20} - \)\(35\!\cdots\!18\)\( p^{21} T^{21} + 4895660342061304707 p^{24} T^{22} - 3842038823167800 p^{27} T^{23} + 4769157286117 p^{30} T^{24} - 2797052904 p^{33} T^{25} + 3107635 p^{36} T^{26} - 1026 p^{39} T^{27} + p^{42} T^{28} \)
79 \( 1 + 695 T + 4044286 T^{2} + 3465725820 T^{3} + 8596244038646 T^{4} + 7940032953450455 T^{5} + 12559364654469057256 T^{6} + \)\(11\!\cdots\!55\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} + \)\(11\!\cdots\!75\)\( T^{9} + \)\(11\!\cdots\!73\)\( T^{10} + \)\(94\!\cdots\!60\)\( T^{11} + \)\(81\!\cdots\!28\)\( T^{12} + \)\(58\!\cdots\!10\)\( T^{13} + \)\(44\!\cdots\!58\)\( T^{14} + \)\(58\!\cdots\!10\)\( p^{3} T^{15} + \)\(81\!\cdots\!28\)\( p^{6} T^{16} + \)\(94\!\cdots\!60\)\( p^{9} T^{17} + \)\(11\!\cdots\!73\)\( p^{12} T^{18} + \)\(11\!\cdots\!75\)\( p^{15} T^{19} + \)\(13\!\cdots\!41\)\( p^{18} T^{20} + \)\(11\!\cdots\!55\)\( p^{21} T^{21} + 12559364654469057256 p^{24} T^{22} + 7940032953450455 p^{27} T^{23} + 8596244038646 p^{30} T^{24} + 3465725820 p^{33} T^{25} + 4044286 p^{36} T^{26} + 695 p^{39} T^{27} + p^{42} T^{28} \)
83 \( 1 + 5315 T + 16409309 T^{2} + 35892059959 T^{3} + 61982119976375 T^{4} + 1064014654509184 p T^{5} + \)\(10\!\cdots\!86\)\( T^{6} + \)\(11\!\cdots\!86\)\( T^{7} + \)\(10\!\cdots\!52\)\( T^{8} + \)\(79\!\cdots\!50\)\( T^{9} + \)\(51\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!64\)\( T^{11} + \)\(89\!\cdots\!88\)\( T^{12} - \)\(11\!\cdots\!68\)\( T^{13} - \)\(20\!\cdots\!30\)\( T^{14} - \)\(11\!\cdots\!68\)\( p^{3} T^{15} + \)\(89\!\cdots\!88\)\( p^{6} T^{16} + \)\(26\!\cdots\!64\)\( p^{9} T^{17} + \)\(51\!\cdots\!74\)\( p^{12} T^{18} + \)\(79\!\cdots\!50\)\( p^{15} T^{19} + \)\(10\!\cdots\!52\)\( p^{18} T^{20} + \)\(11\!\cdots\!86\)\( p^{21} T^{21} + \)\(10\!\cdots\!86\)\( p^{24} T^{22} + 1064014654509184 p^{28} T^{23} + 61982119976375 p^{30} T^{24} + 35892059959 p^{33} T^{25} + 16409309 p^{36} T^{26} + 5315 p^{39} T^{27} + p^{42} T^{28} \)
89 \( 1 + 16 p T + 4789776 T^{2} + 5461708598 T^{3} + 11527078653061 T^{4} + 11643035761958456 T^{5} + 19466059480618971937 T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(25\!\cdots\!38\)\( T^{8} + \)\(21\!\cdots\!44\)\( T^{9} + \)\(27\!\cdots\!51\)\( T^{10} + \)\(21\!\cdots\!86\)\( T^{11} + \)\(24\!\cdots\!37\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(18\!\cdots\!58\)\( T^{14} + \)\(17\!\cdots\!28\)\( p^{3} T^{15} + \)\(24\!\cdots\!37\)\( p^{6} T^{16} + \)\(21\!\cdots\!86\)\( p^{9} T^{17} + \)\(27\!\cdots\!51\)\( p^{12} T^{18} + \)\(21\!\cdots\!44\)\( p^{15} T^{19} + \)\(25\!\cdots\!38\)\( p^{18} T^{20} + \)\(17\!\cdots\!60\)\( p^{21} T^{21} + 19466059480618971937 p^{24} T^{22} + 11643035761958456 p^{27} T^{23} + 11527078653061 p^{30} T^{24} + 5461708598 p^{33} T^{25} + 4789776 p^{36} T^{26} + 16 p^{40} T^{27} + p^{42} T^{28} \)
97 \( 1 + 3 p T + 5338238 T^{2} + 2798287510 T^{3} + 14644437955701 T^{4} + 9094940429450768 T^{5} + 29081042552942820935 T^{6} + \)\(17\!\cdots\!66\)\( T^{7} + \)\(45\!\cdots\!96\)\( T^{8} + \)\(26\!\cdots\!63\)\( T^{9} + \)\(60\!\cdots\!62\)\( T^{10} + \)\(34\!\cdots\!53\)\( T^{11} + \)\(67\!\cdots\!22\)\( T^{12} + \)\(36\!\cdots\!28\)\( T^{13} + \)\(66\!\cdots\!97\)\( T^{14} + \)\(36\!\cdots\!28\)\( p^{3} T^{15} + \)\(67\!\cdots\!22\)\( p^{6} T^{16} + \)\(34\!\cdots\!53\)\( p^{9} T^{17} + \)\(60\!\cdots\!62\)\( p^{12} T^{18} + \)\(26\!\cdots\!63\)\( p^{15} T^{19} + \)\(45\!\cdots\!96\)\( p^{18} T^{20} + \)\(17\!\cdots\!66\)\( p^{21} T^{21} + 29081042552942820935 p^{24} T^{22} + 9094940429450768 p^{27} T^{23} + 14644437955701 p^{30} T^{24} + 2798287510 p^{33} T^{25} + 5338238 p^{36} T^{26} + 3 p^{40} T^{27} + p^{42} T^{28} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.62455088865493123261445204678, −2.61321557164591802991307964904, −2.53298457950116853387509092200, −2.52623925432536407614447650917, −2.49917853992084527121904364546, −2.35352474780712372888292473109, −2.33878725462349952121558559136, −2.25030664919541813849766625440, −2.16136858015930916408380123616, −2.12364901214428686901333695515, −1.91502751001159855911024172179, −1.88125335525693691736704703775, −1.86023140571303293097269545125, −1.82301116344535834634666120481, −1.75013796939215166861839237784, −1.42202897673161948304178206609, −1.41691003589184874533265176849, −1.35265997760158039914150633527, −1.29859625929464557418016292868, −1.23534025218651217113299305588, −1.22031648274512305475006836792, −1.14416233404640394625026866907, −1.13830233171167748025671801699, −1.10933655632223883562419887699, −1.03255193792129941695394175201, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.03255193792129941695394175201, 1.10933655632223883562419887699, 1.13830233171167748025671801699, 1.14416233404640394625026866907, 1.22031648274512305475006836792, 1.23534025218651217113299305588, 1.29859625929464557418016292868, 1.35265997760158039914150633527, 1.41691003589184874533265176849, 1.42202897673161948304178206609, 1.75013796939215166861839237784, 1.82301116344535834634666120481, 1.86023140571303293097269545125, 1.88125335525693691736704703775, 1.91502751001159855911024172179, 2.12364901214428686901333695515, 2.16136858015930916408380123616, 2.25030664919541813849766625440, 2.33878725462349952121558559136, 2.35352474780712372888292473109, 2.49917853992084527121904364546, 2.52623925432536407614447650917, 2.53298457950116853387509092200, 2.61321557164591802991307964904, 2.62455088865493123261445204678

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

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