Properties

Degree $28$
Conductor $7.206\times 10^{30}$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·3-s + 42·5-s + 66·7-s + 50·9-s − 414·13-s + 420·15-s + 1.22e3·17-s + 5.67e3·19-s + 660·21-s + 2.90e3·23-s − 1.35e3·25-s + 254·27-s + 2.77e3·35-s − 1.79e3·37-s − 4.14e3·39-s + 1.16e4·41-s − 3.98e3·43-s + 2.10e3·45-s − 1.27e3·47-s + 2.17e3·49-s + 1.22e4·51-s + 5.88e3·53-s + 5.67e4·57-s − 8.50e3·59-s + 2.05e4·61-s + 3.30e3·63-s − 1.73e4·65-s + ⋯
L(s)  = 1  + 0.641·3-s + 0.751·5-s + 0.509·7-s + 0.205·9-s − 0.679·13-s + 0.481·15-s + 1.02·17-s + 3.60·19-s + 0.326·21-s + 1.14·23-s − 0.432·25-s + 0.0670·27-s + 0.382·35-s − 0.214·37-s − 0.435·39-s + 1.08·41-s − 0.328·43-s + 0.154·45-s − 0.0843·47-s + 0.129·49-s + 0.657·51-s + 0.287·53-s + 2.31·57-s − 0.318·59-s + 0.707·61-s + 0.104·63-s − 0.510·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{70} \cdot 5^{14}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{70} \cdot 5^{14}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{160} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{70} \cdot 5^{14} ,\ ( \ : [5/2]^{14} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(9.265857704\)
\(L(\frac12)\) \(\approx\) \(9.265857704\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 42 T + 623 p T^{2} - 15532 p T^{3} - 64003 p^{2} T^{4} + 24826 p^{4} T^{5} - 5225877 p^{5} T^{6} + 11846872 p^{7} T^{7} - 5225877 p^{10} T^{8} + 24826 p^{14} T^{9} - 64003 p^{17} T^{10} - 15532 p^{21} T^{11} + 623 p^{26} T^{12} - 42 p^{30} T^{13} + p^{35} T^{14} \)
good3 \( 1 - 10 T + 50 T^{2} - 254 T^{3} - 91369 T^{4} + 1354100 T^{5} - 8940292 T^{6} + 156709276 T^{7} + 3134936461 T^{8} - 12907304306 p T^{9} + 32288446142 p^{2} T^{10} - 27718466434 p^{4} T^{11} + 116231885755 p^{4} T^{12} - 1947008143736 p^{5} T^{13} + 26412951129032 p^{6} T^{14} - 1947008143736 p^{10} T^{15} + 116231885755 p^{14} T^{16} - 27718466434 p^{19} T^{17} + 32288446142 p^{22} T^{18} - 12907304306 p^{26} T^{19} + 3134936461 p^{30} T^{20} + 156709276 p^{35} T^{21} - 8940292 p^{40} T^{22} + 1354100 p^{45} T^{23} - 91369 p^{50} T^{24} - 254 p^{55} T^{25} + 50 p^{60} T^{26} - 10 p^{65} T^{27} + p^{70} T^{28} \)
7 \( 1 - 66 T + 2178 T^{2} + 3104786 T^{3} - 7972673 p^{2} T^{4} - 28598040732 T^{5} + 7558178349116 T^{6} - 730117525814980 T^{7} - 100452887411613603 T^{8} + 13091475719468548578 T^{9} + 64214231979007274450 p T^{10} - \)\(38\!\cdots\!06\)\( T^{11} + \)\(37\!\cdots\!31\)\( T^{12} + \)\(47\!\cdots\!84\)\( T^{13} - \)\(77\!\cdots\!76\)\( T^{14} + \)\(47\!\cdots\!84\)\( p^{5} T^{15} + \)\(37\!\cdots\!31\)\( p^{10} T^{16} - \)\(38\!\cdots\!06\)\( p^{15} T^{17} + 64214231979007274450 p^{21} T^{18} + 13091475719468548578 p^{25} T^{19} - 100452887411613603 p^{30} T^{20} - 730117525814980 p^{35} T^{21} + 7558178349116 p^{40} T^{22} - 28598040732 p^{45} T^{23} - 7972673 p^{52} T^{24} + 3104786 p^{55} T^{25} + 2178 p^{60} T^{26} - 66 p^{65} T^{27} + p^{70} T^{28} \)
11 \( 1 - 137970 p T^{2} + 1118279059571 T^{4} - 531239593921187868 T^{6} + \)\(18\!\cdots\!61\)\( T^{8} - \)\(48\!\cdots\!18\)\( T^{10} + \)\(10\!\cdots\!67\)\( T^{12} - \)\(18\!\cdots\!88\)\( T^{14} + \)\(10\!\cdots\!67\)\( p^{10} T^{16} - \)\(48\!\cdots\!18\)\( p^{20} T^{18} + \)\(18\!\cdots\!61\)\( p^{30} T^{20} - 531239593921187868 p^{40} T^{22} + 1118279059571 p^{50} T^{24} - 137970 p^{61} T^{26} + p^{70} T^{28} \)
13 \( 1 + 414 T + 85698 T^{2} + 485371686 T^{3} + 407510115667 T^{4} - 25330305055892 T^{5} + 72383288599671444 T^{6} + \)\(13\!\cdots\!64\)\( T^{7} + \)\(19\!\cdots\!85\)\( T^{8} - \)\(26\!\cdots\!74\)\( T^{9} + \)\(24\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!02\)\( T^{11} - \)\(59\!\cdots\!93\)\( p^{2} T^{12} - \)\(21\!\cdots\!88\)\( p T^{13} + \)\(44\!\cdots\!64\)\( T^{14} - \)\(21\!\cdots\!88\)\( p^{6} T^{15} - \)\(59\!\cdots\!93\)\( p^{12} T^{16} + \)\(12\!\cdots\!02\)\( p^{15} T^{17} + \)\(24\!\cdots\!22\)\( p^{20} T^{18} - \)\(26\!\cdots\!74\)\( p^{25} T^{19} + \)\(19\!\cdots\!85\)\( p^{30} T^{20} + \)\(13\!\cdots\!64\)\( p^{35} T^{21} + 72383288599671444 p^{40} T^{22} - 25330305055892 p^{45} T^{23} + 407510115667 p^{50} T^{24} + 485371686 p^{55} T^{25} + 85698 p^{60} T^{26} + 414 p^{65} T^{27} + p^{70} T^{28} \)
17 \( 1 - 1222 T + 746642 T^{2} - 1023656774 T^{3} - 7517818418661 T^{4} + 9805314295428420 T^{5} - 5845038493789599340 T^{6} + \)\(85\!\cdots\!80\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(32\!\cdots\!70\)\( T^{9} + \)\(19\!\cdots\!70\)\( T^{10} - \)\(33\!\cdots\!50\)\( T^{11} - \)\(20\!\cdots\!37\)\( T^{12} + \)\(66\!\cdots\!24\)\( T^{13} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(66\!\cdots\!24\)\( p^{5} T^{15} - \)\(20\!\cdots\!37\)\( p^{10} T^{16} - \)\(33\!\cdots\!50\)\( p^{15} T^{17} + \)\(19\!\cdots\!70\)\( p^{20} T^{18} - \)\(32\!\cdots\!70\)\( p^{25} T^{19} + \)\(21\!\cdots\!05\)\( p^{30} T^{20} + \)\(85\!\cdots\!80\)\( p^{35} T^{21} - 5845038493789599340 p^{40} T^{22} + 9805314295428420 p^{45} T^{23} - 7517818418661 p^{50} T^{24} - 1023656774 p^{55} T^{25} + 746642 p^{60} T^{26} - 1222 p^{65} T^{27} + p^{70} T^{28} \)
19 \( ( 1 - 2836 T + 12192485 T^{2} - 21773970856 T^{3} + 3075663164351 p T^{4} - 77435325554759820 T^{5} + \)\(17\!\cdots\!85\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!85\)\( p^{5} T^{8} - 77435325554759820 p^{10} T^{9} + 3075663164351 p^{16} T^{10} - 21773970856 p^{20} T^{11} + 12192485 p^{25} T^{12} - 2836 p^{30} T^{13} + p^{35} T^{14} )^{2} \)
23 \( 1 - 2902 T + 4210802 T^{2} + 8585545830 T^{3} - 77137172452561 T^{4} + 5298422918089420 T^{5} + \)\(34\!\cdots\!32\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} + \)\(43\!\cdots\!33\)\( T^{8} + \)\(14\!\cdots\!50\)\( T^{9} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(57\!\cdots\!98\)\( T^{11} - \)\(19\!\cdots\!21\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!40\)\( T^{14} - \)\(51\!\cdots\!28\)\( p^{5} T^{15} - \)\(19\!\cdots\!21\)\( p^{10} T^{16} + \)\(57\!\cdots\!98\)\( p^{15} T^{17} - \)\(10\!\cdots\!54\)\( p^{20} T^{18} + \)\(14\!\cdots\!50\)\( p^{25} T^{19} + \)\(43\!\cdots\!33\)\( p^{30} T^{20} - \)\(17\!\cdots\!48\)\( p^{35} T^{21} + \)\(34\!\cdots\!32\)\( p^{40} T^{22} + 5298422918089420 p^{45} T^{23} - 77137172452561 p^{50} T^{24} + 8585545830 p^{55} T^{25} + 4210802 p^{60} T^{26} - 2902 p^{65} T^{27} + p^{70} T^{28} \)
29 \( 1 - 127109206 T^{2} + 8400089254893139 T^{4} - \)\(38\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!02\)\( T^{10} + \)\(98\!\cdots\!23\)\( T^{12} - \)\(21\!\cdots\!92\)\( T^{14} + \)\(98\!\cdots\!23\)\( p^{10} T^{16} - \)\(39\!\cdots\!02\)\( p^{20} T^{18} + \)\(13\!\cdots\!57\)\( p^{30} T^{20} - \)\(38\!\cdots\!76\)\( p^{40} T^{22} + 8400089254893139 p^{50} T^{24} - 127109206 p^{60} T^{26} + p^{70} T^{28} \)
31 \( 1 - 265679454 T^{2} + 34933996110970539 T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!17\)\( T^{8} - \)\(92\!\cdots\!38\)\( T^{10} + \)\(36\!\cdots\!43\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!43\)\( p^{10} T^{16} - \)\(92\!\cdots\!38\)\( p^{20} T^{18} + \)\(19\!\cdots\!17\)\( p^{30} T^{20} - \)\(30\!\cdots\!64\)\( p^{40} T^{22} + 34933996110970539 p^{50} T^{24} - 265679454 p^{60} T^{26} + p^{70} T^{28} \)
37 \( 1 + 1790 T + 1602050 T^{2} - 1085215470730 T^{3} - 5589943426343837 T^{4} - 5189725886098796788 T^{5} + \)\(58\!\cdots\!80\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!01\)\( T^{8} - \)\(14\!\cdots\!14\)\( T^{9} - \)\(11\!\cdots\!78\)\( T^{10} - \)\(12\!\cdots\!50\)\( T^{11} - \)\(40\!\cdots\!45\)\( T^{12} + \)\(67\!\cdots\!72\)\( T^{13} + \)\(47\!\cdots\!96\)\( T^{14} + \)\(67\!\cdots\!72\)\( p^{5} T^{15} - \)\(40\!\cdots\!45\)\( p^{10} T^{16} - \)\(12\!\cdots\!50\)\( p^{15} T^{17} - \)\(11\!\cdots\!78\)\( p^{20} T^{18} - \)\(14\!\cdots\!14\)\( p^{25} T^{19} + \)\(20\!\cdots\!01\)\( p^{30} T^{20} + \)\(37\!\cdots\!60\)\( p^{35} T^{21} + \)\(58\!\cdots\!80\)\( p^{40} T^{22} - 5189725886098796788 p^{45} T^{23} - 5589943426343837 p^{50} T^{24} - 1085215470730 p^{55} T^{25} + 1602050 p^{60} T^{26} + 1790 p^{65} T^{27} + p^{70} T^{28} \)
41 \( ( 1 - 142 p T + 527400583 T^{2} - 3511555483012 T^{3} + 138605008918596701 T^{4} - \)\(95\!\cdots\!50\)\( T^{5} + \)\(23\!\cdots\!95\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(23\!\cdots\!95\)\( p^{5} T^{8} - \)\(95\!\cdots\!50\)\( p^{10} T^{9} + 138605008918596701 p^{15} T^{10} - 3511555483012 p^{20} T^{11} + 527400583 p^{25} T^{12} - 142 p^{31} T^{13} + p^{35} T^{14} )^{2} \)
43 \( 1 + 3982 T + 7928162 T^{2} + 5090675647610 T^{3} + 44126507369726119 T^{4} + 95654814508201483332 T^{5} + \)\(12\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!12\)\( T^{7} + \)\(46\!\cdots\!93\)\( T^{8} + \)\(33\!\cdots\!78\)\( T^{9} + \)\(35\!\cdots\!38\)\( T^{10} + \)\(10\!\cdots\!70\)\( T^{11} + \)\(78\!\cdots\!59\)\( T^{12} + \)\(84\!\cdots\!64\)\( T^{13} + \)\(18\!\cdots\!08\)\( T^{14} + \)\(84\!\cdots\!64\)\( p^{5} T^{15} + \)\(78\!\cdots\!59\)\( p^{10} T^{16} + \)\(10\!\cdots\!70\)\( p^{15} T^{17} + \)\(35\!\cdots\!38\)\( p^{20} T^{18} + \)\(33\!\cdots\!78\)\( p^{25} T^{19} + \)\(46\!\cdots\!93\)\( p^{30} T^{20} + \)\(15\!\cdots\!12\)\( p^{35} T^{21} + \)\(12\!\cdots\!96\)\( p^{40} T^{22} + 95654814508201483332 p^{45} T^{23} + 44126507369726119 p^{50} T^{24} + 5090675647610 p^{55} T^{25} + 7928162 p^{60} T^{26} + 3982 p^{65} T^{27} + p^{70} T^{28} \)
47 \( 1 + 1278 T + 816642 T^{2} + 351784473730 T^{3} + 13143771115043551 T^{4} - 55735540634425394844 T^{5} - \)\(20\!\cdots\!24\)\( T^{6} - \)\(35\!\cdots\!24\)\( T^{7} + \)\(21\!\cdots\!21\)\( T^{8} + \)\(27\!\cdots\!06\)\( T^{9} + \)\(23\!\cdots\!58\)\( T^{10} + \)\(14\!\cdots\!22\)\( T^{11} - \)\(33\!\cdots\!25\)\( T^{12} - \)\(57\!\cdots\!44\)\( T^{13} - \)\(35\!\cdots\!52\)\( T^{14} - \)\(57\!\cdots\!44\)\( p^{5} T^{15} - \)\(33\!\cdots\!25\)\( p^{10} T^{16} + \)\(14\!\cdots\!22\)\( p^{15} T^{17} + \)\(23\!\cdots\!58\)\( p^{20} T^{18} + \)\(27\!\cdots\!06\)\( p^{25} T^{19} + \)\(21\!\cdots\!21\)\( p^{30} T^{20} - \)\(35\!\cdots\!24\)\( p^{35} T^{21} - \)\(20\!\cdots\!24\)\( p^{40} T^{22} - 55735540634425394844 p^{45} T^{23} + 13143771115043551 p^{50} T^{24} + 351784473730 p^{55} T^{25} + 816642 p^{60} T^{26} + 1278 p^{65} T^{27} + p^{70} T^{28} \)
53 \( 1 - 5882 T + 17298962 T^{2} + 13844528488830 T^{3} - 174311180941592381 T^{4} - \)\(81\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!36\)\( T^{6} - \)\(35\!\cdots\!12\)\( T^{7} - \)\(10\!\cdots\!67\)\( T^{8} + \)\(15\!\cdots\!82\)\( T^{9} + \)\(26\!\cdots\!38\)\( T^{10} - \)\(97\!\cdots\!30\)\( T^{11} + \)\(23\!\cdots\!39\)\( T^{12} + \)\(29\!\cdots\!36\)\( T^{13} - \)\(56\!\cdots\!72\)\( T^{14} + \)\(29\!\cdots\!36\)\( p^{5} T^{15} + \)\(23\!\cdots\!39\)\( p^{10} T^{16} - \)\(97\!\cdots\!30\)\( p^{15} T^{17} + \)\(26\!\cdots\!38\)\( p^{20} T^{18} + \)\(15\!\cdots\!82\)\( p^{25} T^{19} - \)\(10\!\cdots\!67\)\( p^{30} T^{20} - \)\(35\!\cdots\!12\)\( p^{35} T^{21} + \)\(14\!\cdots\!36\)\( p^{40} T^{22} - \)\(81\!\cdots\!52\)\( p^{45} T^{23} - 174311180941592381 p^{50} T^{24} + 13844528488830 p^{55} T^{25} + 17298962 p^{60} T^{26} - 5882 p^{65} T^{27} + p^{70} T^{28} \)
59 \( ( 1 + 4252 T + 2005910877 T^{2} + 20518241068280 T^{3} + 2395790088022200413 T^{4} + \)\(22\!\cdots\!48\)\( T^{5} + \)\(23\!\cdots\!29\)\( T^{6} + \)\(16\!\cdots\!28\)\( T^{7} + \)\(23\!\cdots\!29\)\( p^{5} T^{8} + \)\(22\!\cdots\!48\)\( p^{10} T^{9} + 2395790088022200413 p^{15} T^{10} + 20518241068280 p^{20} T^{11} + 2005910877 p^{25} T^{12} + 4252 p^{30} T^{13} + p^{35} T^{14} )^{2} \)
61 \( ( 1 - 10282 T + 2916253203 T^{2} + 3083320715268 T^{3} + 3046127491767085861 T^{4} + \)\(70\!\cdots\!38\)\( T^{5} + \)\(15\!\cdots\!39\)\( T^{6} + \)\(10\!\cdots\!44\)\( T^{7} + \)\(15\!\cdots\!39\)\( p^{5} T^{8} + \)\(70\!\cdots\!38\)\( p^{10} T^{9} + 3046127491767085861 p^{15} T^{10} + 3083320715268 p^{20} T^{11} + 2916253203 p^{25} T^{12} - 10282 p^{30} T^{13} + p^{35} T^{14} )^{2} \)
67 \( 1 - 107926 T + 5824010738 T^{2} - 286091118212770 T^{3} + 7910959270847152663 T^{4} + \)\(10\!\cdots\!32\)\( T^{5} - \)\(16\!\cdots\!76\)\( T^{6} + \)\(10\!\cdots\!52\)\( T^{7} - \)\(49\!\cdots\!51\)\( T^{8} + \)\(12\!\cdots\!38\)\( T^{9} - \)\(20\!\cdots\!90\)\( T^{10} - \)\(24\!\cdots\!26\)\( T^{11} + \)\(60\!\cdots\!11\)\( T^{12} - \)\(31\!\cdots\!32\)\( T^{13} + \)\(11\!\cdots\!96\)\( T^{14} - \)\(31\!\cdots\!32\)\( p^{5} T^{15} + \)\(60\!\cdots\!11\)\( p^{10} T^{16} - \)\(24\!\cdots\!26\)\( p^{15} T^{17} - \)\(20\!\cdots\!90\)\( p^{20} T^{18} + \)\(12\!\cdots\!38\)\( p^{25} T^{19} - \)\(49\!\cdots\!51\)\( p^{30} T^{20} + \)\(10\!\cdots\!52\)\( p^{35} T^{21} - \)\(16\!\cdots\!76\)\( p^{40} T^{22} + \)\(10\!\cdots\!32\)\( p^{45} T^{23} + 7910959270847152663 p^{50} T^{24} - 286091118212770 p^{55} T^{25} + 5824010738 p^{60} T^{26} - 107926 p^{65} T^{27} + p^{70} T^{28} \)
71 \( 1 - 12494857902 T^{2} + 77600272039468829243 T^{4} - \)\(32\!\cdots\!32\)\( T^{6} + \)\(99\!\cdots\!41\)\( T^{8} - \)\(25\!\cdots\!58\)\( T^{10} + \)\(55\!\cdots\!91\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{14} + \)\(55\!\cdots\!91\)\( p^{10} T^{16} - \)\(25\!\cdots\!58\)\( p^{20} T^{18} + \)\(99\!\cdots\!41\)\( p^{30} T^{20} - \)\(32\!\cdots\!32\)\( p^{40} T^{22} + 77600272039468829243 p^{50} T^{24} - 12494857902 p^{60} T^{26} + p^{70} T^{28} \)
73 \( 1 + 16418 T + 134775362 T^{2} - 257655666175822 T^{3} - 4365114300655601877 T^{4} + \)\(39\!\cdots\!72\)\( T^{5} + \)\(40\!\cdots\!12\)\( T^{6} + \)\(72\!\cdots\!32\)\( T^{7} - \)\(10\!\cdots\!35\)\( T^{8} - \)\(45\!\cdots\!66\)\( T^{9} + \)\(74\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!50\)\( T^{11} + \)\(46\!\cdots\!07\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{13} - \)\(14\!\cdots\!04\)\( T^{14} - \)\(13\!\cdots\!16\)\( p^{5} T^{15} + \)\(46\!\cdots\!07\)\( p^{10} T^{16} + \)\(14\!\cdots\!50\)\( p^{15} T^{17} + \)\(74\!\cdots\!94\)\( p^{20} T^{18} - \)\(45\!\cdots\!66\)\( p^{25} T^{19} - \)\(10\!\cdots\!35\)\( p^{30} T^{20} + \)\(72\!\cdots\!32\)\( p^{35} T^{21} + \)\(40\!\cdots\!12\)\( p^{40} T^{22} + \)\(39\!\cdots\!72\)\( p^{45} T^{23} - 4365114300655601877 p^{50} T^{24} - 257655666175822 p^{55} T^{25} + 134775362 p^{60} T^{26} + 16418 p^{65} T^{27} + p^{70} T^{28} \)
79 \( ( 1 + 73272 T + 14910422889 T^{2} + 713614712327088 T^{3} + 91878682859930077941 T^{4} + \)\(38\!\cdots\!48\)\( p T^{5} + \)\(35\!\cdots\!29\)\( T^{6} + \)\(97\!\cdots\!08\)\( T^{7} + \)\(35\!\cdots\!29\)\( p^{5} T^{8} + \)\(38\!\cdots\!48\)\( p^{11} T^{9} + 91878682859930077941 p^{15} T^{10} + 713614712327088 p^{20} T^{11} + 14910422889 p^{25} T^{12} + 73272 p^{30} T^{13} + p^{35} T^{14} )^{2} \)
83 \( 1 + 36398 T + 662407202 T^{2} - 351755925637942 T^{3} - 18959358682913357257 T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} + \)\(60\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} - \)\(15\!\cdots\!90\)\( T^{9} - \)\(10\!\cdots\!50\)\( T^{10} - \)\(86\!\cdots\!90\)\( T^{11} - \)\(70\!\cdots\!85\)\( T^{12} + \)\(38\!\cdots\!20\)\( T^{13} + \)\(31\!\cdots\!80\)\( T^{14} + \)\(38\!\cdots\!20\)\( p^{5} T^{15} - \)\(70\!\cdots\!85\)\( p^{10} T^{16} - \)\(86\!\cdots\!90\)\( p^{15} T^{17} - \)\(10\!\cdots\!50\)\( p^{20} T^{18} - \)\(15\!\cdots\!90\)\( p^{25} T^{19} + \)\(11\!\cdots\!21\)\( p^{30} T^{20} + \)\(60\!\cdots\!36\)\( p^{35} T^{21} + \)\(11\!\cdots\!08\)\( p^{40} T^{22} + \)\(10\!\cdots\!44\)\( p^{45} T^{23} - 18959358682913357257 p^{50} T^{24} - 351755925637942 p^{55} T^{25} + 662407202 p^{60} T^{26} + 36398 p^{65} T^{27} + p^{70} T^{28} \)
89 \( 1 - 30334038558 T^{2} + \)\(36\!\cdots\!35\)\( T^{4} - \)\(20\!\cdots\!52\)\( T^{6} + \)\(39\!\cdots\!69\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(42\!\cdots\!95\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!95\)\( p^{10} T^{16} - \)\(12\!\cdots\!10\)\( p^{20} T^{18} + \)\(39\!\cdots\!69\)\( p^{30} T^{20} - \)\(20\!\cdots\!52\)\( p^{40} T^{22} + \)\(36\!\cdots\!35\)\( p^{50} T^{24} - 30334038558 p^{60} T^{26} + p^{70} T^{28} \)
97 \( 1 + 60314 T + 1818889298 T^{2} - 623535693824102 T^{3} - \)\(16\!\cdots\!77\)\( T^{4} - \)\(33\!\cdots\!64\)\( T^{5} + \)\(29\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!52\)\( T^{7} + \)\(13\!\cdots\!33\)\( p T^{8} - \)\(44\!\cdots\!50\)\( T^{9} + \)\(18\!\cdots\!50\)\( T^{10} + \)\(91\!\cdots\!50\)\( T^{11} + \)\(44\!\cdots\!75\)\( T^{12} - \)\(82\!\cdots\!00\)\( T^{13} - \)\(73\!\cdots\!00\)\( T^{14} - \)\(82\!\cdots\!00\)\( p^{5} T^{15} + \)\(44\!\cdots\!75\)\( p^{10} T^{16} + \)\(91\!\cdots\!50\)\( p^{15} T^{17} + \)\(18\!\cdots\!50\)\( p^{20} T^{18} - \)\(44\!\cdots\!50\)\( p^{25} T^{19} + \)\(13\!\cdots\!33\)\( p^{31} T^{20} + \)\(51\!\cdots\!52\)\( p^{35} T^{21} + \)\(29\!\cdots\!52\)\( p^{40} T^{22} - \)\(33\!\cdots\!64\)\( p^{45} T^{23} - \)\(16\!\cdots\!77\)\( p^{50} T^{24} - 623535693824102 p^{55} T^{25} + 1818889298 p^{60} T^{26} + 60314 p^{65} T^{27} + p^{70} T^{28} \)
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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

−3.19914015631181393228383475479, −2.89753591575533337412428868412, −2.82907750635253006250112616136, −2.60870918179524455051715736523, −2.57115347528033955768273724667, −2.46320163188241689481934650008, −2.41371282080863118701772723750, −2.32772902192153018731103182030, −2.21796798113967994700151704038, −2.21195161939312623989040451545, −1.80642404621072424911334776632, −1.71961416883680084948431854474, −1.70130479412006016918873077545, −1.48185048105592344971770333785, −1.35857038319010044384302043991, −1.27371045554518024638722051706, −1.17902911520192284746270255018, −0.976546217881951214578717580638, −0.933270648023861040921065627100, −0.930462747598537041568561568035, −0.819059916348767244595269756700, −0.45600610741081031681833466976, −0.34058295467615935608739459195, −0.27544117574174641306384999406, −0.07853856617321036398884190249, 0.07853856617321036398884190249, 0.27544117574174641306384999406, 0.34058295467615935608739459195, 0.45600610741081031681833466976, 0.819059916348767244595269756700, 0.930462747598537041568561568035, 0.933270648023861040921065627100, 0.976546217881951214578717580638, 1.17902911520192284746270255018, 1.27371045554518024638722051706, 1.35857038319010044384302043991, 1.48185048105592344971770333785, 1.70130479412006016918873077545, 1.71961416883680084948431854474, 1.80642404621072424911334776632, 2.21195161939312623989040451545, 2.21796798113967994700151704038, 2.32772902192153018731103182030, 2.41371282080863118701772723750, 2.46320163188241689481934650008, 2.57115347528033955768273724667, 2.60870918179524455051715736523, 2.82907750635253006250112616136, 2.89753591575533337412428868412, 3.19914015631181393228383475479

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

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