Properties

Label 28-38e14-1.1-c7e14-0-0
Degree $28$
Conductor $1.309\times 10^{22}$
Sign $1$
Analytic cond. $1.10314\times 10^{15}$
Root an. cond. $3.44537$
Motivic weight $7$
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  + 56·2-s + 55·3-s + 1.34e3·4-s − 126·5-s + 3.08e3·6-s + 1.92e3·7-s + 1.43e4·8-s + 6.86e3·9-s − 7.05e3·10-s + 646·11-s + 7.39e4·12-s − 1.30e3·13-s + 1.07e5·14-s − 6.93e3·15-s − 5.73e4·16-s + 1.05e4·17-s + 3.84e5·18-s − 8.74e4·19-s − 1.69e5·20-s + 1.06e5·21-s + 3.61e4·22-s − 1.13e5·23-s + 7.88e5·24-s + 2.56e5·25-s − 7.32e4·26-s + 2.95e5·27-s + 2.59e6·28-s + ⋯
L(s)  = 1  + 4.94·2-s + 1.17·3-s + 21/2·4-s − 0.450·5-s + 5.82·6-s + 2.12·7-s + 9.89·8-s + 3.13·9-s − 2.23·10-s + 0.146·11-s + 12.3·12-s − 0.165·13-s + 10.5·14-s − 0.530·15-s − 7/2·16-s + 0.519·17-s + 15.5·18-s − 2.92·19-s − 4.73·20-s + 2.49·21-s + 0.724·22-s − 1.95·23-s + 11.6·24-s + 3.28·25-s − 0.817·26-s + 2.89·27-s + 22.3·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{14} \cdot 19^{14}\)
Sign: $1$
Analytic conductor: \(1.10314\times 10^{15}\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{14} \cdot 19^{14} ,\ ( \ : [7/2]^{14} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(60.32220317\)
\(L(\frac12)\) \(\approx\) \(60.32220317\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{3} T + p^{6} T^{2} )^{7} \)
19 \( 1 + 87463 T + 1218560863 T^{2} - 6964130879772 p T^{3} - 16565864543728164 p^{2} T^{4} - 296111379180295936 p^{4} T^{5} + 81768039604196311040 p^{6} T^{6} + \)\(51\!\cdots\!12\)\( p^{9} T^{7} + 81768039604196311040 p^{13} T^{8} - 296111379180295936 p^{18} T^{9} - 16565864543728164 p^{23} T^{10} - 6964130879772 p^{29} T^{11} + 1218560863 p^{35} T^{12} + 87463 p^{42} T^{13} + p^{49} T^{14} \)
good3 \( 1 - 55 T - 3841 T^{2} + 293156 T^{3} + 3675062 T^{4} - 640984438 T^{5} + 26323137955 T^{6} - 9035802667 p^{2} T^{7} - 11802552205336 p^{2} T^{8} + 2270427153787 p^{7} T^{9} + 466489659918655 p^{5} T^{10} - 19751051619624590 p^{6} T^{11} + 525220387513325633 p^{6} T^{12} + 2407777672290349915 p^{8} T^{13} - \)\(24\!\cdots\!02\)\( p^{8} T^{14} + 2407777672290349915 p^{15} T^{15} + 525220387513325633 p^{20} T^{16} - 19751051619624590 p^{27} T^{17} + 466489659918655 p^{33} T^{18} + 2270427153787 p^{42} T^{19} - 11802552205336 p^{44} T^{20} - 9035802667 p^{51} T^{21} + 26323137955 p^{56} T^{22} - 640984438 p^{63} T^{23} + 3675062 p^{70} T^{24} + 293156 p^{77} T^{25} - 3841 p^{84} T^{26} - 55 p^{91} T^{27} + p^{98} T^{28} \)
5 \( 1 + 126 T - 240428 T^{2} - 27369376 T^{3} + 28299463426 T^{4} + 449736338962 p T^{5} - 14322870951186 p^{3} T^{6} + 15314833737526 p^{5} T^{7} + 12048582895951971 p^{5} T^{8} - 2195807825180397186 p^{6} T^{9} + \)\(30\!\cdots\!12\)\( p^{6} T^{10} + \)\(50\!\cdots\!14\)\( p^{7} T^{11} - \)\(32\!\cdots\!81\)\( p^{9} T^{12} - \)\(76\!\cdots\!62\)\( p^{9} T^{13} + \)\(55\!\cdots\!46\)\( p^{10} T^{14} - \)\(76\!\cdots\!62\)\( p^{16} T^{15} - \)\(32\!\cdots\!81\)\( p^{23} T^{16} + \)\(50\!\cdots\!14\)\( p^{28} T^{17} + \)\(30\!\cdots\!12\)\( p^{34} T^{18} - 2195807825180397186 p^{41} T^{19} + 12048582895951971 p^{47} T^{20} + 15314833737526 p^{54} T^{21} - 14322870951186 p^{59} T^{22} + 449736338962 p^{64} T^{23} + 28299463426 p^{70} T^{24} - 27369376 p^{77} T^{25} - 240428 p^{84} T^{26} + 126 p^{91} T^{27} + p^{98} T^{28} \)
7 \( ( 1 - 964 T + 1872337 T^{2} - 230830232 p T^{3} + 1808662214969 T^{4} - 581011503608780 T^{5} + 106887007875296783 p T^{6} - 43927976075692954640 T^{7} + 106887007875296783 p^{8} T^{8} - 581011503608780 p^{14} T^{9} + 1808662214969 p^{21} T^{10} - 230830232 p^{29} T^{11} + 1872337 p^{35} T^{12} - 964 p^{42} T^{13} + p^{49} T^{14} )^{2} \)
11 \( ( 1 - 323 T + 35272703 T^{2} + 59049498580 T^{3} + 620645521891300 T^{4} + 1484973402846849056 T^{5} + \)\(14\!\cdots\!10\)\( T^{6} + \)\(19\!\cdots\!26\)\( T^{7} + \)\(14\!\cdots\!10\)\( p^{7} T^{8} + 1484973402846849056 p^{14} T^{9} + 620645521891300 p^{21} T^{10} + 59049498580 p^{28} T^{11} + 35272703 p^{35} T^{12} - 323 p^{42} T^{13} + p^{49} T^{14} )^{2} \)
13 \( 1 + 1308 T + 14219084 T^{2} + 31661129308 p T^{3} - 2795323299098726 T^{4} + 11851587983821375892 T^{5} - \)\(26\!\cdots\!90\)\( T^{6} - \)\(96\!\cdots\!64\)\( T^{7} + \)\(14\!\cdots\!47\)\( T^{8} - \)\(96\!\cdots\!04\)\( T^{9} - \)\(50\!\cdots\!60\)\( T^{10} - \)\(22\!\cdots\!20\)\( T^{11} - \)\(16\!\cdots\!65\)\( T^{12} + \)\(16\!\cdots\!80\)\( T^{13} - \)\(23\!\cdots\!38\)\( T^{14} + \)\(16\!\cdots\!80\)\( p^{7} T^{15} - \)\(16\!\cdots\!65\)\( p^{14} T^{16} - \)\(22\!\cdots\!20\)\( p^{21} T^{17} - \)\(50\!\cdots\!60\)\( p^{28} T^{18} - \)\(96\!\cdots\!04\)\( p^{35} T^{19} + \)\(14\!\cdots\!47\)\( p^{42} T^{20} - \)\(96\!\cdots\!64\)\( p^{49} T^{21} - \)\(26\!\cdots\!90\)\( p^{56} T^{22} + 11851587983821375892 p^{63} T^{23} - 2795323299098726 p^{70} T^{24} + 31661129308 p^{78} T^{25} + 14219084 p^{84} T^{26} + 1308 p^{91} T^{27} + p^{98} T^{28} \)
17 \( 1 - 10528 T - 765279628 T^{2} + 13866949754092 T^{3} + 93590884203881794 T^{4} - \)\(41\!\cdots\!08\)\( T^{5} + \)\(82\!\cdots\!98\)\( T^{6} - \)\(18\!\cdots\!56\)\( T^{7} - \)\(17\!\cdots\!25\)\( T^{8} + \)\(87\!\cdots\!48\)\( T^{9} - \)\(80\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!16\)\( T^{11} - \)\(58\!\cdots\!53\)\( T^{12} - \)\(11\!\cdots\!24\)\( T^{13} + \)\(64\!\cdots\!18\)\( T^{14} - \)\(11\!\cdots\!24\)\( p^{7} T^{15} - \)\(58\!\cdots\!53\)\( p^{14} T^{16} + \)\(24\!\cdots\!16\)\( p^{21} T^{17} - \)\(80\!\cdots\!16\)\( p^{28} T^{18} + \)\(87\!\cdots\!48\)\( p^{35} T^{19} - \)\(17\!\cdots\!25\)\( p^{42} T^{20} - \)\(18\!\cdots\!56\)\( p^{49} T^{21} + \)\(82\!\cdots\!98\)\( p^{56} T^{22} - \)\(41\!\cdots\!08\)\( p^{63} T^{23} + 93590884203881794 p^{70} T^{24} + 13866949754092 p^{77} T^{25} - 765279628 p^{84} T^{26} - 10528 p^{91} T^{27} + p^{98} T^{28} \)
23 \( 1 + 113822 T - 2000886916 T^{2} - 373831550598132 T^{3} + 5359450061941137686 T^{4} - \)\(11\!\cdots\!34\)\( T^{5} - \)\(14\!\cdots\!54\)\( T^{6} + \)\(57\!\cdots\!94\)\( T^{7} + \)\(42\!\cdots\!43\)\( T^{8} + \)\(25\!\cdots\!94\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} + \)\(12\!\cdots\!50\)\( T^{11} + \)\(33\!\cdots\!79\)\( T^{12} - \)\(16\!\cdots\!82\)\( T^{13} - \)\(36\!\cdots\!58\)\( T^{14} - \)\(16\!\cdots\!82\)\( p^{7} T^{15} + \)\(33\!\cdots\!79\)\( p^{14} T^{16} + \)\(12\!\cdots\!50\)\( p^{21} T^{17} + \)\(54\!\cdots\!76\)\( p^{28} T^{18} + \)\(25\!\cdots\!94\)\( p^{35} T^{19} + \)\(42\!\cdots\!43\)\( p^{42} T^{20} + \)\(57\!\cdots\!94\)\( p^{49} T^{21} - \)\(14\!\cdots\!54\)\( p^{56} T^{22} - \)\(11\!\cdots\!34\)\( p^{63} T^{23} + 5359450061941137686 p^{70} T^{24} - 373831550598132 p^{77} T^{25} - 2000886916 p^{84} T^{26} + 113822 p^{91} T^{27} + p^{98} T^{28} \)
29 \( 1 - 167706 T - 49993022276 T^{2} + 4730414346909376 T^{3} + \)\(20\!\cdots\!50\)\( T^{4} - \)\(43\!\cdots\!38\)\( T^{5} - \)\(49\!\cdots\!38\)\( T^{6} - \)\(24\!\cdots\!18\)\( T^{7} + \)\(85\!\cdots\!67\)\( T^{8} + \)\(88\!\cdots\!22\)\( T^{9} - \)\(61\!\cdots\!52\)\( T^{10} - \)\(16\!\cdots\!02\)\( T^{11} - \)\(62\!\cdots\!09\)\( T^{12} + \)\(11\!\cdots\!82\)\( T^{13} + \)\(27\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!82\)\( p^{7} T^{15} - \)\(62\!\cdots\!09\)\( p^{14} T^{16} - \)\(16\!\cdots\!02\)\( p^{21} T^{17} - \)\(61\!\cdots\!52\)\( p^{28} T^{18} + \)\(88\!\cdots\!22\)\( p^{35} T^{19} + \)\(85\!\cdots\!67\)\( p^{42} T^{20} - \)\(24\!\cdots\!18\)\( p^{49} T^{21} - \)\(49\!\cdots\!38\)\( p^{56} T^{22} - \)\(43\!\cdots\!38\)\( p^{63} T^{23} + \)\(20\!\cdots\!50\)\( p^{70} T^{24} + 4730414346909376 p^{77} T^{25} - 49993022276 p^{84} T^{26} - 167706 p^{91} T^{27} + p^{98} T^{28} \)
31 \( ( 1 - 100890 T + 119900944537 T^{2} - 9963144467980740 T^{3} + \)\(69\!\cdots\!29\)\( T^{4} - \)\(51\!\cdots\!70\)\( T^{5} + \)\(26\!\cdots\!85\)\( T^{6} - \)\(17\!\cdots\!56\)\( T^{7} + \)\(26\!\cdots\!85\)\( p^{7} T^{8} - \)\(51\!\cdots\!70\)\( p^{14} T^{9} + \)\(69\!\cdots\!29\)\( p^{21} T^{10} - 9963144467980740 p^{28} T^{11} + 119900944537 p^{35} T^{12} - 100890 p^{42} T^{13} + p^{49} T^{14} )^{2} \)
37 \( ( 1 - 190462 T + 364636942663 T^{2} - 1334232639434676 p T^{3} + \)\(63\!\cdots\!01\)\( T^{4} - \)\(66\!\cdots\!90\)\( T^{5} + \)\(76\!\cdots\!87\)\( T^{6} - \)\(69\!\cdots\!40\)\( T^{7} + \)\(76\!\cdots\!87\)\( p^{7} T^{8} - \)\(66\!\cdots\!90\)\( p^{14} T^{9} + \)\(63\!\cdots\!01\)\( p^{21} T^{10} - 1334232639434676 p^{29} T^{11} + 364636942663 p^{35} T^{12} - 190462 p^{42} T^{13} + p^{49} T^{14} )^{2} \)
41 \( 1 - 66301 T - 603724117711 T^{2} + 16645519220623550 T^{3} + \)\(14\!\cdots\!56\)\( T^{4} + \)\(92\!\cdots\!40\)\( T^{5} - \)\(22\!\cdots\!99\)\( T^{6} - \)\(66\!\cdots\!23\)\( T^{7} + \)\(42\!\cdots\!60\)\( T^{8} + \)\(19\!\cdots\!91\)\( T^{9} - \)\(80\!\cdots\!45\)\( T^{10} - \)\(31\!\cdots\!36\)\( T^{11} + \)\(58\!\cdots\!59\)\( T^{12} + \)\(20\!\cdots\!59\)\( T^{13} + \)\(30\!\cdots\!42\)\( T^{14} + \)\(20\!\cdots\!59\)\( p^{7} T^{15} + \)\(58\!\cdots\!59\)\( p^{14} T^{16} - \)\(31\!\cdots\!36\)\( p^{21} T^{17} - \)\(80\!\cdots\!45\)\( p^{28} T^{18} + \)\(19\!\cdots\!91\)\( p^{35} T^{19} + \)\(42\!\cdots\!60\)\( p^{42} T^{20} - \)\(66\!\cdots\!23\)\( p^{49} T^{21} - \)\(22\!\cdots\!99\)\( p^{56} T^{22} + \)\(92\!\cdots\!40\)\( p^{63} T^{23} + \)\(14\!\cdots\!56\)\( p^{70} T^{24} + 16645519220623550 p^{77} T^{25} - 603724117711 p^{84} T^{26} - 66301 p^{91} T^{27} + p^{98} T^{28} \)
43 \( 1 - 726564 T - 923581956088 T^{2} + 855538102544616312 T^{3} + \)\(38\!\cdots\!10\)\( T^{4} - \)\(47\!\cdots\!92\)\( T^{5} - \)\(12\!\cdots\!98\)\( T^{6} + \)\(17\!\cdots\!44\)\( T^{7} + \)\(45\!\cdots\!83\)\( T^{8} - \)\(50\!\cdots\!88\)\( T^{9} - \)\(15\!\cdots\!96\)\( T^{10} + \)\(11\!\cdots\!60\)\( T^{11} + \)\(41\!\cdots\!63\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} - \)\(10\!\cdots\!26\)\( T^{14} - \)\(13\!\cdots\!56\)\( p^{7} T^{15} + \)\(41\!\cdots\!63\)\( p^{14} T^{16} + \)\(11\!\cdots\!60\)\( p^{21} T^{17} - \)\(15\!\cdots\!96\)\( p^{28} T^{18} - \)\(50\!\cdots\!88\)\( p^{35} T^{19} + \)\(45\!\cdots\!83\)\( p^{42} T^{20} + \)\(17\!\cdots\!44\)\( p^{49} T^{21} - \)\(12\!\cdots\!98\)\( p^{56} T^{22} - \)\(47\!\cdots\!92\)\( p^{63} T^{23} + \)\(38\!\cdots\!10\)\( p^{70} T^{24} + 855538102544616312 p^{77} T^{25} - 923581956088 p^{84} T^{26} - 726564 p^{91} T^{27} + p^{98} T^{28} \)
47 \( 1 + 2373788 T + 1370507061080 T^{2} - 261963874590034752 T^{3} + \)\(41\!\cdots\!14\)\( T^{4} + \)\(90\!\cdots\!96\)\( T^{5} + \)\(16\!\cdots\!42\)\( T^{6} + \)\(33\!\cdots\!16\)\( T^{7} + \)\(62\!\cdots\!79\)\( T^{8} + \)\(12\!\cdots\!72\)\( T^{9} - \)\(23\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!84\)\( T^{11} + \)\(44\!\cdots\!87\)\( T^{12} + \)\(42\!\cdots\!24\)\( T^{13} + \)\(72\!\cdots\!06\)\( T^{14} + \)\(42\!\cdots\!24\)\( p^{7} T^{15} + \)\(44\!\cdots\!87\)\( p^{14} T^{16} + \)\(11\!\cdots\!84\)\( p^{21} T^{17} - \)\(23\!\cdots\!88\)\( p^{28} T^{18} + \)\(12\!\cdots\!72\)\( p^{35} T^{19} + \)\(62\!\cdots\!79\)\( p^{42} T^{20} + \)\(33\!\cdots\!16\)\( p^{49} T^{21} + \)\(16\!\cdots\!42\)\( p^{56} T^{22} + \)\(90\!\cdots\!96\)\( p^{63} T^{23} + \)\(41\!\cdots\!14\)\( p^{70} T^{24} - 261963874590034752 p^{77} T^{25} + 1370507061080 p^{84} T^{26} + 2373788 p^{91} T^{27} + p^{98} T^{28} \)
53 \( 1 - 161792 T - 5012387856388 T^{2} + 1826276832278361028 T^{3} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(68\!\cdots\!76\)\( T^{5} - \)\(22\!\cdots\!50\)\( T^{6} + \)\(15\!\cdots\!68\)\( T^{7} + \)\(24\!\cdots\!23\)\( T^{8} - \)\(24\!\cdots\!88\)\( T^{9} - \)\(12\!\cdots\!32\)\( T^{10} + \)\(24\!\cdots\!04\)\( T^{11} - \)\(77\!\cdots\!25\)\( T^{12} - \)\(22\!\cdots\!12\)\( p T^{13} + \)\(75\!\cdots\!26\)\( p^{2} T^{14} - \)\(22\!\cdots\!12\)\( p^{8} T^{15} - \)\(77\!\cdots\!25\)\( p^{14} T^{16} + \)\(24\!\cdots\!04\)\( p^{21} T^{17} - \)\(12\!\cdots\!32\)\( p^{28} T^{18} - \)\(24\!\cdots\!88\)\( p^{35} T^{19} + \)\(24\!\cdots\!23\)\( p^{42} T^{20} + \)\(15\!\cdots\!68\)\( p^{49} T^{21} - \)\(22\!\cdots\!50\)\( p^{56} T^{22} - \)\(68\!\cdots\!76\)\( p^{63} T^{23} + \)\(13\!\cdots\!86\)\( p^{70} T^{24} + 1826276832278361028 p^{77} T^{25} - 5012387856388 p^{84} T^{26} - 161792 p^{91} T^{27} + p^{98} T^{28} \)
59 \( 1 + 1845767 T - 11341077687337 T^{2} - 14324535590263921004 T^{3} + \)\(83\!\cdots\!30\)\( T^{4} + \)\(61\!\cdots\!86\)\( T^{5} - \)\(44\!\cdots\!13\)\( T^{6} - \)\(19\!\cdots\!61\)\( T^{7} + \)\(17\!\cdots\!00\)\( T^{8} + \)\(43\!\cdots\!99\)\( T^{9} - \)\(61\!\cdots\!59\)\( T^{10} - \)\(79\!\cdots\!54\)\( T^{11} + \)\(18\!\cdots\!81\)\( T^{12} + \)\(80\!\cdots\!53\)\( T^{13} - \)\(48\!\cdots\!98\)\( T^{14} + \)\(80\!\cdots\!53\)\( p^{7} T^{15} + \)\(18\!\cdots\!81\)\( p^{14} T^{16} - \)\(79\!\cdots\!54\)\( p^{21} T^{17} - \)\(61\!\cdots\!59\)\( p^{28} T^{18} + \)\(43\!\cdots\!99\)\( p^{35} T^{19} + \)\(17\!\cdots\!00\)\( p^{42} T^{20} - \)\(19\!\cdots\!61\)\( p^{49} T^{21} - \)\(44\!\cdots\!13\)\( p^{56} T^{22} + \)\(61\!\cdots\!86\)\( p^{63} T^{23} + \)\(83\!\cdots\!30\)\( p^{70} T^{24} - 14324535590263921004 p^{77} T^{25} - 11341077687337 p^{84} T^{26} + 1845767 p^{91} T^{27} + p^{98} T^{28} \)
61 \( 1 + 1600418 T - 9134366904084 T^{2} + 765879370215665640 T^{3} + \)\(59\!\cdots\!86\)\( T^{4} - \)\(64\!\cdots\!54\)\( T^{5} - \)\(16\!\cdots\!38\)\( T^{6} + \)\(61\!\cdots\!66\)\( p T^{7} + \)\(67\!\cdots\!59\)\( T^{8} - \)\(59\!\cdots\!18\)\( T^{9} + \)\(24\!\cdots\!28\)\( T^{10} - \)\(15\!\cdots\!70\)\( T^{11} + \)\(44\!\cdots\!79\)\( T^{12} + \)\(45\!\cdots\!78\)\( T^{13} - \)\(26\!\cdots\!66\)\( T^{14} + \)\(45\!\cdots\!78\)\( p^{7} T^{15} + \)\(44\!\cdots\!79\)\( p^{14} T^{16} - \)\(15\!\cdots\!70\)\( p^{21} T^{17} + \)\(24\!\cdots\!28\)\( p^{28} T^{18} - \)\(59\!\cdots\!18\)\( p^{35} T^{19} + \)\(67\!\cdots\!59\)\( p^{42} T^{20} + \)\(61\!\cdots\!66\)\( p^{50} T^{21} - \)\(16\!\cdots\!38\)\( p^{56} T^{22} - \)\(64\!\cdots\!54\)\( p^{63} T^{23} + \)\(59\!\cdots\!86\)\( p^{70} T^{24} + 765879370215665640 p^{77} T^{25} - 9134366904084 p^{84} T^{26} + 1600418 p^{91} T^{27} + p^{98} T^{28} \)
67 \( 1 + 8911929 T + 21649850201615 T^{2} + 2818391344931485652 T^{3} + \)\(57\!\cdots\!90\)\( T^{4} + \)\(22\!\cdots\!54\)\( T^{5} - \)\(15\!\cdots\!33\)\( T^{6} - \)\(69\!\cdots\!31\)\( T^{7} - \)\(71\!\cdots\!72\)\( T^{8} - \)\(11\!\cdots\!67\)\( T^{9} - \)\(81\!\cdots\!31\)\( T^{10} - \)\(45\!\cdots\!74\)\( T^{11} + \)\(70\!\cdots\!69\)\( T^{12} + \)\(14\!\cdots\!83\)\( T^{13} + \)\(11\!\cdots\!06\)\( T^{14} + \)\(14\!\cdots\!83\)\( p^{7} T^{15} + \)\(70\!\cdots\!69\)\( p^{14} T^{16} - \)\(45\!\cdots\!74\)\( p^{21} T^{17} - \)\(81\!\cdots\!31\)\( p^{28} T^{18} - \)\(11\!\cdots\!67\)\( p^{35} T^{19} - \)\(71\!\cdots\!72\)\( p^{42} T^{20} - \)\(69\!\cdots\!31\)\( p^{49} T^{21} - \)\(15\!\cdots\!33\)\( p^{56} T^{22} + \)\(22\!\cdots\!54\)\( p^{63} T^{23} + \)\(57\!\cdots\!90\)\( p^{70} T^{24} + 2818391344931485652 p^{77} T^{25} + 21649850201615 p^{84} T^{26} + 8911929 p^{91} T^{27} + p^{98} T^{28} \)
71 \( 1 - 517154 T - 37070234195980 T^{2} - 7109081551216262324 T^{3} + \)\(61\!\cdots\!10\)\( T^{4} + \)\(46\!\cdots\!38\)\( T^{5} - \)\(84\!\cdots\!70\)\( T^{6} - \)\(29\!\cdots\!26\)\( T^{7} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(10\!\cdots\!46\)\( T^{9} - \)\(15\!\cdots\!68\)\( T^{10} - \)\(35\!\cdots\!14\)\( T^{11} + \)\(15\!\cdots\!47\)\( T^{12} + \)\(38\!\cdots\!50\)\( T^{13} - \)\(14\!\cdots\!86\)\( T^{14} + \)\(38\!\cdots\!50\)\( p^{7} T^{15} + \)\(15\!\cdots\!47\)\( p^{14} T^{16} - \)\(35\!\cdots\!14\)\( p^{21} T^{17} - \)\(15\!\cdots\!68\)\( p^{28} T^{18} - \)\(10\!\cdots\!46\)\( p^{35} T^{19} + \)\(12\!\cdots\!07\)\( p^{42} T^{20} - \)\(29\!\cdots\!26\)\( p^{49} T^{21} - \)\(84\!\cdots\!70\)\( p^{56} T^{22} + \)\(46\!\cdots\!38\)\( p^{63} T^{23} + \)\(61\!\cdots\!10\)\( p^{70} T^{24} - 7109081551216262324 p^{77} T^{25} - 37070234195980 p^{84} T^{26} - 517154 p^{91} T^{27} + p^{98} T^{28} \)
73 \( 1 + 9558049 T + 9940048244901 T^{2} - \)\(12\!\cdots\!58\)\( T^{3} - \)\(19\!\cdots\!36\)\( T^{4} + \)\(18\!\cdots\!56\)\( T^{5} - \)\(20\!\cdots\!11\)\( T^{6} - \)\(27\!\cdots\!85\)\( T^{7} - \)\(59\!\cdots\!28\)\( T^{8} + \)\(14\!\cdots\!17\)\( T^{9} - \)\(12\!\cdots\!89\)\( T^{10} - \)\(10\!\cdots\!12\)\( T^{11} + \)\(15\!\cdots\!71\)\( T^{12} + \)\(14\!\cdots\!53\)\( T^{13} + \)\(51\!\cdots\!34\)\( T^{14} + \)\(14\!\cdots\!53\)\( p^{7} T^{15} + \)\(15\!\cdots\!71\)\( p^{14} T^{16} - \)\(10\!\cdots\!12\)\( p^{21} T^{17} - \)\(12\!\cdots\!89\)\( p^{28} T^{18} + \)\(14\!\cdots\!17\)\( p^{35} T^{19} - \)\(59\!\cdots\!28\)\( p^{42} T^{20} - \)\(27\!\cdots\!85\)\( p^{49} T^{21} - \)\(20\!\cdots\!11\)\( p^{56} T^{22} + \)\(18\!\cdots\!56\)\( p^{63} T^{23} - \)\(19\!\cdots\!36\)\( p^{70} T^{24} - \)\(12\!\cdots\!58\)\( p^{77} T^{25} + 9940048244901 p^{84} T^{26} + 9558049 p^{91} T^{27} + p^{98} T^{28} \)
79 \( 1 - 7963520 T - 15171701371008 T^{2} + \)\(21\!\cdots\!40\)\( T^{3} - \)\(11\!\cdots\!62\)\( T^{4} - \)\(71\!\cdots\!56\)\( T^{5} - \)\(55\!\cdots\!58\)\( T^{6} + \)\(27\!\cdots\!48\)\( T^{7} + \)\(57\!\cdots\!51\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{9} + \)\(64\!\cdots\!72\)\( T^{10} + \)\(19\!\cdots\!68\)\( T^{11} - \)\(74\!\cdots\!61\)\( T^{12} + \)\(18\!\cdots\!64\)\( T^{13} - \)\(60\!\cdots\!74\)\( T^{14} + \)\(18\!\cdots\!64\)\( p^{7} T^{15} - \)\(74\!\cdots\!61\)\( p^{14} T^{16} + \)\(19\!\cdots\!68\)\( p^{21} T^{17} + \)\(64\!\cdots\!72\)\( p^{28} T^{18} - \)\(18\!\cdots\!48\)\( p^{35} T^{19} + \)\(57\!\cdots\!51\)\( p^{42} T^{20} + \)\(27\!\cdots\!48\)\( p^{49} T^{21} - \)\(55\!\cdots\!58\)\( p^{56} T^{22} - \)\(71\!\cdots\!56\)\( p^{63} T^{23} - \)\(11\!\cdots\!62\)\( p^{70} T^{24} + \)\(21\!\cdots\!40\)\( p^{77} T^{25} - 15171701371008 p^{84} T^{26} - 7963520 p^{91} T^{27} + p^{98} T^{28} \)
83 \( ( 1 - 15308443 T + 212048914964135 T^{2} - \)\(18\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!32\)\( T^{4} - \)\(95\!\cdots\!92\)\( T^{5} + \)\(60\!\cdots\!90\)\( T^{6} - \)\(31\!\cdots\!50\)\( T^{7} + \)\(60\!\cdots\!90\)\( p^{7} T^{8} - \)\(95\!\cdots\!92\)\( p^{14} T^{9} + \)\(15\!\cdots\!32\)\( p^{21} T^{10} - \)\(18\!\cdots\!36\)\( p^{28} T^{11} + 212048914964135 p^{35} T^{12} - 15308443 p^{42} T^{13} + p^{49} T^{14} )^{2} \)
89 \( 1 - 11829436 T - 73165976926900 T^{2} + \)\(60\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(33\!\cdots\!24\)\( T^{5} - \)\(55\!\cdots\!26\)\( T^{6} - \)\(20\!\cdots\!16\)\( T^{7} + \)\(27\!\cdots\!87\)\( T^{8} + \)\(16\!\cdots\!68\)\( T^{9} + \)\(39\!\cdots\!64\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} - \)\(63\!\cdots\!05\)\( T^{12} + \)\(14\!\cdots\!20\)\( T^{13} + \)\(50\!\cdots\!78\)\( T^{14} + \)\(14\!\cdots\!20\)\( p^{7} T^{15} - \)\(63\!\cdots\!05\)\( p^{14} T^{16} - \)\(12\!\cdots\!68\)\( p^{21} T^{17} + \)\(39\!\cdots\!64\)\( p^{28} T^{18} + \)\(16\!\cdots\!68\)\( p^{35} T^{19} + \)\(27\!\cdots\!87\)\( p^{42} T^{20} - \)\(20\!\cdots\!16\)\( p^{49} T^{21} - \)\(55\!\cdots\!26\)\( p^{56} T^{22} - \)\(33\!\cdots\!24\)\( p^{63} T^{23} + \)\(10\!\cdots\!38\)\( p^{70} T^{24} + \)\(60\!\cdots\!00\)\( p^{77} T^{25} - 73165976926900 p^{84} T^{26} - 11829436 p^{91} T^{27} + p^{98} T^{28} \)
97 \( 1 + 8033723 T - 287993815511247 T^{2} - \)\(43\!\cdots\!06\)\( T^{3} + \)\(35\!\cdots\!92\)\( T^{4} + \)\(10\!\cdots\!52\)\( T^{5} - \)\(33\!\cdots\!23\)\( T^{6} - \)\(14\!\cdots\!31\)\( T^{7} - \)\(62\!\cdots\!48\)\( T^{8} + \)\(14\!\cdots\!79\)\( T^{9} + \)\(12\!\cdots\!19\)\( T^{10} - \)\(95\!\cdots\!96\)\( T^{11} - \)\(15\!\cdots\!05\)\( T^{12} + \)\(29\!\cdots\!75\)\( T^{13} + \)\(14\!\cdots\!94\)\( T^{14} + \)\(29\!\cdots\!75\)\( p^{7} T^{15} - \)\(15\!\cdots\!05\)\( p^{14} T^{16} - \)\(95\!\cdots\!96\)\( p^{21} T^{17} + \)\(12\!\cdots\!19\)\( p^{28} T^{18} + \)\(14\!\cdots\!79\)\( p^{35} T^{19} - \)\(62\!\cdots\!48\)\( p^{42} T^{20} - \)\(14\!\cdots\!31\)\( p^{49} T^{21} - \)\(33\!\cdots\!23\)\( p^{56} T^{22} + \)\(10\!\cdots\!52\)\( p^{63} T^{23} + \)\(35\!\cdots\!92\)\( p^{70} T^{24} - \)\(43\!\cdots\!06\)\( p^{77} T^{25} - 287993815511247 p^{84} T^{26} + 8033723 p^{91} T^{27} + p^{98} T^{28} \)
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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

−4.23186210577492710035796356683, −3.88220364418299905856719058912, −3.78054407288141869424675129659, −3.57327190302978587379892849356, −3.48140071023841265487026098957, −3.40351744961117569383598240696, −3.31185577405350268675449380372, −3.16888310036975452016936884718, −2.91052626785445048864886218792, −2.74680093844361424052457815283, −2.60575885286186021840494555319, −2.59283830430939613623150339755, −2.37453771521164405897726090966, −2.11318374335025961698933099866, −1.99601157744214582847555339045, −1.90490011749950398119113337842, −1.60327645031020513536435969366, −1.44521307340803983067578243129, −1.32399168913822680483637590012, −1.09782176147166385029670770638, −1.09414621904567048106726702246, −0.845731616728194843741934547155, −0.35287574146464979864726285681, −0.32972433904153613300330676778, −0.11289476345704231023888973995, 0.11289476345704231023888973995, 0.32972433904153613300330676778, 0.35287574146464979864726285681, 0.845731616728194843741934547155, 1.09414621904567048106726702246, 1.09782176147166385029670770638, 1.32399168913822680483637590012, 1.44521307340803983067578243129, 1.60327645031020513536435969366, 1.90490011749950398119113337842, 1.99601157744214582847555339045, 2.11318374335025961698933099866, 2.37453771521164405897726090966, 2.59283830430939613623150339755, 2.60575885286186021840494555319, 2.74680093844361424052457815283, 2.91052626785445048864886218792, 3.16888310036975452016936884718, 3.31185577405350268675449380372, 3.40351744961117569383598240696, 3.48140071023841265487026098957, 3.57327190302978587379892849356, 3.78054407288141869424675129659, 3.88220364418299905856719058912, 4.23186210577492710035796356683

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

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