Properties

Label 22-29e11-1.1-c11e11-0-0
Degree $22$
Conductor $1.220\times 10^{16}$
Sign $-1$
Analytic cond. $6.72180\times 10^{14}$
Root an. cond. $4.72037$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s − 982·3-s − 6.17e3·4-s − 2.74e3·5-s + 3.14e4·6-s − 4.94e4·7-s + 1.80e5·8-s − 3.26e5·9-s + 8.76e4·10-s − 6.12e5·11-s + 6.06e6·12-s + 1.51e6·13-s + 1.58e6·14-s + 2.69e6·15-s + 1.59e7·16-s − 3.29e6·17-s + 1.04e7·18-s − 4.41e7·19-s + 1.69e7·20-s + 4.85e7·21-s + 1.95e7·22-s − 8.86e7·23-s − 1.77e8·24-s − 2.86e8·25-s − 4.83e7·26-s + 6.15e8·27-s + 3.05e8·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.33·3-s − 3.01·4-s − 0.392·5-s + 1.64·6-s − 1.11·7-s + 1.94·8-s − 1.84·9-s + 0.277·10-s − 1.14·11-s + 7.03·12-s + 1.12·13-s + 0.786·14-s + 0.914·15-s + 3.80·16-s − 0.562·17-s + 1.30·18-s − 4.08·19-s + 1.18·20-s + 2.59·21-s + 0.810·22-s − 2.87·23-s − 4.54·24-s − 5.87·25-s − 0.797·26-s + 8.25·27-s + 3.35·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{11}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(29^{11}\)
Sign: $-1$
Analytic conductor: \(6.72180\times 10^{14}\)
Root analytic conductor: \(4.72037\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{29} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 29^{11} ,\ ( \ : [11/2]^{11} ),\ -1 )\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( ( 1 - p^{5} T )^{11} \)
good2 \( 1 + p^{5} T + 7203 T^{2} + 123873 p T^{3} + 3835287 p^{3} T^{4} + 71672295 p^{4} T^{5} + 6105687227 p^{4} T^{6} + 115160263473 p^{5} T^{7} + 251563179131 p^{10} T^{8} + 2396833475121 p^{12} T^{9} + 4442367867211 p^{17} T^{10} + 165190638739169 p^{17} T^{11} + 4442367867211 p^{28} T^{12} + 2396833475121 p^{34} T^{13} + 251563179131 p^{43} T^{14} + 115160263473 p^{49} T^{15} + 6105687227 p^{59} T^{16} + 71672295 p^{70} T^{17} + 3835287 p^{80} T^{18} + 123873 p^{89} T^{19} + 7203 p^{99} T^{20} + p^{115} T^{21} + p^{121} T^{22} \)
3 \( 1 + 982 T + 1291096 T^{2} + 324327620 p T^{3} + 89249639390 p^{2} T^{4} + 2089151773942 p^{5} T^{5} + 446309199272755 p^{6} T^{6} + 27258267053314912 p^{8} T^{7} + 1630667769104635255 p^{10} T^{8} + 87928874031617437918 p^{12} T^{9} + \)\(45\!\cdots\!18\)\( p^{14} T^{10} + \)\(21\!\cdots\!48\)\( p^{16} T^{11} + \)\(45\!\cdots\!18\)\( p^{25} T^{12} + 87928874031617437918 p^{34} T^{13} + 1630667769104635255 p^{43} T^{14} + 27258267053314912 p^{52} T^{15} + 446309199272755 p^{61} T^{16} + 2089151773942 p^{71} T^{17} + 89249639390 p^{79} T^{18} + 324327620 p^{89} T^{19} + 1291096 p^{99} T^{20} + 982 p^{110} T^{21} + p^{121} T^{22} \)
5 \( 1 + 548 p T + 294406248 T^{2} + 178254573474 p T^{3} + 1733576644485454 p^{2} T^{4} + 1208706558999821428 p^{3} T^{5} + \)\(67\!\cdots\!81\)\( p^{4} T^{6} + \)\(53\!\cdots\!28\)\( p^{5} T^{7} + \)\(19\!\cdots\!59\)\( p^{6} T^{8} + \)\(16\!\cdots\!64\)\( p^{7} T^{9} + \)\(47\!\cdots\!58\)\( p^{8} T^{10} + \)\(38\!\cdots\!06\)\( p^{9} T^{11} + \)\(47\!\cdots\!58\)\( p^{19} T^{12} + \)\(16\!\cdots\!64\)\( p^{29} T^{13} + \)\(19\!\cdots\!59\)\( p^{39} T^{14} + \)\(53\!\cdots\!28\)\( p^{49} T^{15} + \)\(67\!\cdots\!81\)\( p^{59} T^{16} + 1208706558999821428 p^{69} T^{17} + 1733576644485454 p^{79} T^{18} + 178254573474 p^{89} T^{19} + 294406248 p^{99} T^{20} + 548 p^{111} T^{21} + p^{121} T^{22} \)
7 \( 1 + 49432 T + 9750797397 T^{2} + 505614040906832 T^{3} + 47645025668223072143 T^{4} + \)\(22\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!47\)\( T^{6} + \)\(62\!\cdots\!20\)\( T^{7} + \)\(50\!\cdots\!50\)\( p T^{8} + \)\(37\!\cdots\!92\)\( p^{3} T^{9} + \)\(19\!\cdots\!02\)\( p^{3} T^{10} + \)\(10\!\cdots\!28\)\( p^{4} T^{11} + \)\(19\!\cdots\!02\)\( p^{14} T^{12} + \)\(37\!\cdots\!92\)\( p^{25} T^{13} + \)\(50\!\cdots\!50\)\( p^{34} T^{14} + \)\(62\!\cdots\!20\)\( p^{44} T^{15} + \)\(15\!\cdots\!47\)\( p^{55} T^{16} + \)\(22\!\cdots\!00\)\( p^{66} T^{17} + 47645025668223072143 p^{77} T^{18} + 505614040906832 p^{88} T^{19} + 9750797397 p^{99} T^{20} + 49432 p^{110} T^{21} + p^{121} T^{22} \)
11 \( 1 + 612246 T + 1648047744584 T^{2} + 90721387769922444 p T^{3} + \)\(13\!\cdots\!50\)\( T^{4} + \)\(80\!\cdots\!74\)\( T^{5} + \)\(79\!\cdots\!87\)\( T^{6} + \)\(43\!\cdots\!24\)\( T^{7} + \)\(31\!\cdots\!41\)\( p T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(12\!\cdots\!94\)\( T^{10} + \)\(56\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!94\)\( p^{11} T^{12} + \)\(17\!\cdots\!86\)\( p^{22} T^{13} + \)\(31\!\cdots\!41\)\( p^{34} T^{14} + \)\(43\!\cdots\!24\)\( p^{44} T^{15} + \)\(79\!\cdots\!87\)\( p^{55} T^{16} + \)\(80\!\cdots\!74\)\( p^{66} T^{17} + \)\(13\!\cdots\!50\)\( p^{77} T^{18} + 90721387769922444 p^{89} T^{19} + 1648047744584 p^{99} T^{20} + 612246 p^{110} T^{21} + p^{121} T^{22} \)
13 \( 1 - 1510364 T + 9967969194928 T^{2} - 13366250069482232846 T^{3} + \)\(40\!\cdots\!50\)\( p T^{4} - \)\(64\!\cdots\!48\)\( T^{5} + \)\(19\!\cdots\!89\)\( T^{6} - \)\(16\!\cdots\!72\)\( p T^{7} + \)\(53\!\cdots\!95\)\( T^{8} - \)\(55\!\cdots\!40\)\( T^{9} + \)\(89\!\cdots\!34\)\( p T^{10} - \)\(66\!\cdots\!58\)\( p^{2} T^{11} + \)\(89\!\cdots\!34\)\( p^{12} T^{12} - \)\(55\!\cdots\!40\)\( p^{22} T^{13} + \)\(53\!\cdots\!95\)\( p^{33} T^{14} - \)\(16\!\cdots\!72\)\( p^{45} T^{15} + \)\(19\!\cdots\!89\)\( p^{55} T^{16} - \)\(64\!\cdots\!48\)\( p^{66} T^{17} + \)\(40\!\cdots\!50\)\( p^{78} T^{18} - 13366250069482232846 p^{88} T^{19} + 9967969194928 p^{99} T^{20} - 1510364 p^{110} T^{21} + p^{121} T^{22} \)
17 \( 1 + 193594 p T + 224855458314079 T^{2} + \)\(96\!\cdots\!44\)\( T^{3} + \)\(24\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!62\)\( T^{5} + \)\(62\!\cdots\!09\)\( p^{2} T^{6} + \)\(10\!\cdots\!84\)\( T^{7} + \)\(96\!\cdots\!62\)\( T^{8} + \)\(62\!\cdots\!32\)\( T^{9} + \)\(40\!\cdots\!38\)\( T^{10} + \)\(25\!\cdots\!32\)\( T^{11} + \)\(40\!\cdots\!38\)\( p^{11} T^{12} + \)\(62\!\cdots\!32\)\( p^{22} T^{13} + \)\(96\!\cdots\!62\)\( p^{33} T^{14} + \)\(10\!\cdots\!84\)\( p^{44} T^{15} + \)\(62\!\cdots\!09\)\( p^{57} T^{16} + \)\(13\!\cdots\!62\)\( p^{66} T^{17} + \)\(24\!\cdots\!95\)\( p^{77} T^{18} + \)\(96\!\cdots\!44\)\( p^{88} T^{19} + 224855458314079 p^{99} T^{20} + 193594 p^{111} T^{21} + p^{121} T^{22} \)
19 \( 1 + 44121388 T + 1660524581848749 T^{2} + \)\(41\!\cdots\!64\)\( T^{3} + \)\(95\!\cdots\!31\)\( T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(30\!\cdots\!15\)\( T^{6} + \)\(46\!\cdots\!72\)\( T^{7} + \)\(67\!\cdots\!22\)\( T^{8} + \)\(85\!\cdots\!88\)\( T^{9} + \)\(55\!\cdots\!46\)\( p T^{10} + \)\(32\!\cdots\!36\)\( p^{2} T^{11} + \)\(55\!\cdots\!46\)\( p^{12} T^{12} + \)\(85\!\cdots\!88\)\( p^{22} T^{13} + \)\(67\!\cdots\!22\)\( p^{33} T^{14} + \)\(46\!\cdots\!72\)\( p^{44} T^{15} + \)\(30\!\cdots\!15\)\( p^{55} T^{16} + \)\(17\!\cdots\!96\)\( p^{66} T^{17} + \)\(95\!\cdots\!31\)\( p^{77} T^{18} + \)\(41\!\cdots\!64\)\( p^{88} T^{19} + 1660524581848749 p^{99} T^{20} + 44121388 p^{110} T^{21} + p^{121} T^{22} \)
23 \( 1 + 88684076 T + 8505937460800545 T^{2} + \)\(54\!\cdots\!68\)\( T^{3} + \)\(32\!\cdots\!31\)\( T^{4} + \)\(16\!\cdots\!88\)\( T^{5} + \)\(78\!\cdots\!87\)\( T^{6} + \)\(33\!\cdots\!96\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(48\!\cdots\!56\)\( T^{9} + \)\(16\!\cdots\!62\)\( T^{10} + \)\(53\!\cdots\!44\)\( T^{11} + \)\(16\!\cdots\!62\)\( p^{11} T^{12} + \)\(48\!\cdots\!56\)\( p^{22} T^{13} + \)\(13\!\cdots\!50\)\( p^{33} T^{14} + \)\(33\!\cdots\!96\)\( p^{44} T^{15} + \)\(78\!\cdots\!87\)\( p^{55} T^{16} + \)\(16\!\cdots\!88\)\( p^{66} T^{17} + \)\(32\!\cdots\!31\)\( p^{77} T^{18} + \)\(54\!\cdots\!68\)\( p^{88} T^{19} + 8505937460800545 p^{99} T^{20} + 88684076 p^{110} T^{21} + p^{121} T^{22} \)
31 \( 1 + 292235934 T + 194304895678525668 T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!26\)\( p T^{4} + \)\(44\!\cdots\!22\)\( T^{5} + \)\(11\!\cdots\!03\)\( T^{6} + \)\(24\!\cdots\!60\)\( T^{7} + \)\(51\!\cdots\!43\)\( T^{8} + \)\(96\!\cdots\!94\)\( T^{9} + \)\(17\!\cdots\!98\)\( T^{10} + \)\(28\!\cdots\!68\)\( T^{11} + \)\(17\!\cdots\!98\)\( p^{11} T^{12} + \)\(96\!\cdots\!94\)\( p^{22} T^{13} + \)\(51\!\cdots\!43\)\( p^{33} T^{14} + \)\(24\!\cdots\!60\)\( p^{44} T^{15} + \)\(11\!\cdots\!03\)\( p^{55} T^{16} + \)\(44\!\cdots\!22\)\( p^{66} T^{17} + \)\(59\!\cdots\!26\)\( p^{78} T^{18} + \)\(52\!\cdots\!80\)\( p^{88} T^{19} + 194304895678525668 p^{99} T^{20} + 292235934 p^{110} T^{21} + p^{121} T^{22} \)
37 \( 1 + 1380429338 T + 1922462734570760255 T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!83\)\( T^{4} + \)\(96\!\cdots\!50\)\( T^{5} + \)\(62\!\cdots\!57\)\( T^{6} + \)\(35\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!30\)\( T^{8} + \)\(94\!\cdots\!08\)\( T^{9} + \)\(44\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!08\)\( T^{11} + \)\(44\!\cdots\!98\)\( p^{11} T^{12} + \)\(94\!\cdots\!08\)\( p^{22} T^{13} + \)\(19\!\cdots\!30\)\( p^{33} T^{14} + \)\(35\!\cdots\!80\)\( p^{44} T^{15} + \)\(62\!\cdots\!57\)\( p^{55} T^{16} + \)\(96\!\cdots\!50\)\( p^{66} T^{17} + \)\(14\!\cdots\!83\)\( p^{77} T^{18} + \)\(16\!\cdots\!20\)\( p^{88} T^{19} + 1922462734570760255 p^{99} T^{20} + 1380429338 p^{110} T^{21} + p^{121} T^{22} \)
41 \( 1 + 1062067494 T + 4017275435598161499 T^{2} + \)\(34\!\cdots\!68\)\( T^{3} + \)\(76\!\cdots\!19\)\( T^{4} + \)\(57\!\cdots\!46\)\( T^{5} + \)\(94\!\cdots\!41\)\( T^{6} + \)\(61\!\cdots\!24\)\( T^{7} + \)\(85\!\cdots\!42\)\( T^{8} + \)\(48\!\cdots\!64\)\( T^{9} + \)\(58\!\cdots\!94\)\( T^{10} + \)\(29\!\cdots\!56\)\( T^{11} + \)\(58\!\cdots\!94\)\( p^{11} T^{12} + \)\(48\!\cdots\!64\)\( p^{22} T^{13} + \)\(85\!\cdots\!42\)\( p^{33} T^{14} + \)\(61\!\cdots\!24\)\( p^{44} T^{15} + \)\(94\!\cdots\!41\)\( p^{55} T^{16} + \)\(57\!\cdots\!46\)\( p^{66} T^{17} + \)\(76\!\cdots\!19\)\( p^{77} T^{18} + \)\(34\!\cdots\!68\)\( p^{88} T^{19} + 4017275435598161499 p^{99} T^{20} + 1062067494 p^{110} T^{21} + p^{121} T^{22} \)
43 \( 1 - 74588594 T + 3893459358883448784 T^{2} + \)\(30\!\cdots\!48\)\( T^{3} + \)\(79\!\cdots\!34\)\( T^{4} + \)\(18\!\cdots\!34\)\( T^{5} + \)\(12\!\cdots\!79\)\( T^{6} + \)\(38\!\cdots\!28\)\( T^{7} + \)\(15\!\cdots\!75\)\( T^{8} + \)\(53\!\cdots\!18\)\( T^{9} + \)\(16\!\cdots\!46\)\( T^{10} + \)\(55\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!46\)\( p^{11} T^{12} + \)\(53\!\cdots\!18\)\( p^{22} T^{13} + \)\(15\!\cdots\!75\)\( p^{33} T^{14} + \)\(38\!\cdots\!28\)\( p^{44} T^{15} + \)\(12\!\cdots\!79\)\( p^{55} T^{16} + \)\(18\!\cdots\!34\)\( p^{66} T^{17} + \)\(79\!\cdots\!34\)\( p^{77} T^{18} + \)\(30\!\cdots\!48\)\( p^{88} T^{19} + 3893459358883448784 p^{99} T^{20} - 74588594 p^{110} T^{21} + p^{121} T^{22} \)
47 \( 1 + 1821239394 T + 14950238589906766644 T^{2} + \)\(30\!\cdots\!24\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} + \)\(24\!\cdots\!34\)\( T^{5} + \)\(68\!\cdots\!03\)\( T^{6} + \)\(12\!\cdots\!16\)\( T^{7} + \)\(28\!\cdots\!51\)\( T^{8} + \)\(47\!\cdots\!94\)\( T^{9} + \)\(91\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!76\)\( T^{11} + \)\(91\!\cdots\!10\)\( p^{11} T^{12} + \)\(47\!\cdots\!94\)\( p^{22} T^{13} + \)\(28\!\cdots\!51\)\( p^{33} T^{14} + \)\(12\!\cdots\!16\)\( p^{44} T^{15} + \)\(68\!\cdots\!03\)\( p^{55} T^{16} + \)\(24\!\cdots\!34\)\( p^{66} T^{17} + \)\(12\!\cdots\!02\)\( p^{77} T^{18} + \)\(30\!\cdots\!24\)\( p^{88} T^{19} + 14950238589906766644 p^{99} T^{20} + 1821239394 p^{110} T^{21} + p^{121} T^{22} \)
53 \( 1 - 7818635688 T + 87463804511478195840 T^{2} - \)\(45\!\cdots\!90\)\( T^{3} + \)\(28\!\cdots\!30\)\( T^{4} - \)\(10\!\cdots\!64\)\( T^{5} + \)\(43\!\cdots\!85\)\( T^{6} - \)\(95\!\cdots\!84\)\( T^{7} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(85\!\cdots\!48\)\( T^{9} + \)\(33\!\cdots\!22\)\( T^{10} + \)\(44\!\cdots\!86\)\( T^{11} + \)\(33\!\cdots\!22\)\( p^{11} T^{12} - \)\(85\!\cdots\!48\)\( p^{22} T^{13} + \)\(29\!\cdots\!55\)\( p^{33} T^{14} - \)\(95\!\cdots\!84\)\( p^{44} T^{15} + \)\(43\!\cdots\!85\)\( p^{55} T^{16} - \)\(10\!\cdots\!64\)\( p^{66} T^{17} + \)\(28\!\cdots\!30\)\( p^{77} T^{18} - \)\(45\!\cdots\!90\)\( p^{88} T^{19} + 87463804511478195840 p^{99} T^{20} - 7818635688 p^{110} T^{21} + p^{121} T^{22} \)
59 \( 1 - 1230002712 T + \)\(14\!\cdots\!33\)\( T^{2} - \)\(27\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!07\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(59\!\cdots\!59\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} + \)\(25\!\cdots\!98\)\( T^{8} - \)\(76\!\cdots\!84\)\( T^{9} + \)\(94\!\cdots\!94\)\( T^{10} - \)\(25\!\cdots\!52\)\( T^{11} + \)\(94\!\cdots\!94\)\( p^{11} T^{12} - \)\(76\!\cdots\!84\)\( p^{22} T^{13} + \)\(25\!\cdots\!98\)\( p^{33} T^{14} - \)\(17\!\cdots\!48\)\( p^{44} T^{15} + \)\(59\!\cdots\!59\)\( p^{55} T^{16} - \)\(29\!\cdots\!08\)\( p^{66} T^{17} + \)\(11\!\cdots\!07\)\( p^{77} T^{18} - \)\(27\!\cdots\!92\)\( p^{88} T^{19} + \)\(14\!\cdots\!33\)\( p^{99} T^{20} - 1230002712 p^{110} T^{21} + p^{121} T^{22} \)
61 \( 1 + 18602654230 T + \)\(39\!\cdots\!03\)\( T^{2} + \)\(57\!\cdots\!88\)\( T^{3} + \)\(75\!\cdots\!71\)\( T^{4} + \)\(87\!\cdots\!26\)\( T^{5} + \)\(90\!\cdots\!05\)\( T^{6} + \)\(85\!\cdots\!56\)\( T^{7} + \)\(73\!\cdots\!02\)\( T^{8} + \)\(58\!\cdots\!92\)\( T^{9} + \)\(43\!\cdots\!50\)\( T^{10} + \)\(29\!\cdots\!48\)\( T^{11} + \)\(43\!\cdots\!50\)\( p^{11} T^{12} + \)\(58\!\cdots\!92\)\( p^{22} T^{13} + \)\(73\!\cdots\!02\)\( p^{33} T^{14} + \)\(85\!\cdots\!56\)\( p^{44} T^{15} + \)\(90\!\cdots\!05\)\( p^{55} T^{16} + \)\(87\!\cdots\!26\)\( p^{66} T^{17} + \)\(75\!\cdots\!71\)\( p^{77} T^{18} + \)\(57\!\cdots\!88\)\( p^{88} T^{19} + \)\(39\!\cdots\!03\)\( p^{99} T^{20} + 18602654230 p^{110} T^{21} + p^{121} T^{22} \)
67 \( 1 - 27481284652 T + \)\(10\!\cdots\!41\)\( T^{2} - \)\(22\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!03\)\( T^{4} - \)\(88\!\cdots\!84\)\( T^{5} + \)\(15\!\cdots\!35\)\( T^{6} - \)\(22\!\cdots\!72\)\( T^{7} + \)\(32\!\cdots\!98\)\( T^{8} - \)\(41\!\cdots\!92\)\( T^{9} + \)\(51\!\cdots\!86\)\( T^{10} - \)\(58\!\cdots\!04\)\( T^{11} + \)\(51\!\cdots\!86\)\( p^{11} T^{12} - \)\(41\!\cdots\!92\)\( p^{22} T^{13} + \)\(32\!\cdots\!98\)\( p^{33} T^{14} - \)\(22\!\cdots\!72\)\( p^{44} T^{15} + \)\(15\!\cdots\!35\)\( p^{55} T^{16} - \)\(88\!\cdots\!84\)\( p^{66} T^{17} + \)\(52\!\cdots\!03\)\( p^{77} T^{18} - \)\(22\!\cdots\!60\)\( p^{88} T^{19} + \)\(10\!\cdots\!41\)\( p^{99} T^{20} - 27481284652 p^{110} T^{21} + p^{121} T^{22} \)
71 \( 1 + 20347168516 T + \)\(83\!\cdots\!85\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(49\!\cdots\!87\)\( T^{6} + \)\(71\!\cdots\!88\)\( T^{7} + \)\(15\!\cdots\!02\)\( T^{8} + \)\(98\!\cdots\!68\)\( T^{9} + \)\(56\!\cdots\!06\)\( T^{10} + \)\(41\!\cdots\!16\)\( T^{11} + \)\(56\!\cdots\!06\)\( p^{11} T^{12} + \)\(98\!\cdots\!68\)\( p^{22} T^{13} + \)\(15\!\cdots\!02\)\( p^{33} T^{14} + \)\(71\!\cdots\!88\)\( p^{44} T^{15} + \)\(49\!\cdots\!87\)\( p^{55} T^{16} + \)\(11\!\cdots\!16\)\( p^{66} T^{17} + \)\(24\!\cdots\!79\)\( p^{77} T^{18} + \)\(93\!\cdots\!00\)\( p^{88} T^{19} + \)\(83\!\cdots\!85\)\( p^{99} T^{20} + 20347168516 p^{110} T^{21} + p^{121} T^{22} \)
73 \( 1 + 57740010478 T + \)\(32\!\cdots\!63\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(42\!\cdots\!39\)\( T^{4} + \)\(11\!\cdots\!34\)\( T^{5} + \)\(32\!\cdots\!37\)\( T^{6} + \)\(76\!\cdots\!88\)\( T^{7} + \)\(17\!\cdots\!86\)\( T^{8} + \)\(36\!\cdots\!32\)\( T^{9} + \)\(71\!\cdots\!78\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{11} + \)\(71\!\cdots\!78\)\( p^{11} T^{12} + \)\(36\!\cdots\!32\)\( p^{22} T^{13} + \)\(17\!\cdots\!86\)\( p^{33} T^{14} + \)\(76\!\cdots\!88\)\( p^{44} T^{15} + \)\(32\!\cdots\!37\)\( p^{55} T^{16} + \)\(11\!\cdots\!34\)\( p^{66} T^{17} + \)\(42\!\cdots\!39\)\( p^{77} T^{18} + \)\(12\!\cdots\!12\)\( p^{88} T^{19} + \)\(32\!\cdots\!63\)\( p^{99} T^{20} + 57740010478 p^{110} T^{21} + p^{121} T^{22} \)
79 \( 1 + 120245016462 T + \)\(10\!\cdots\!36\)\( T^{2} + \)\(59\!\cdots\!36\)\( T^{3} + \)\(29\!\cdots\!62\)\( T^{4} + \)\(11\!\cdots\!58\)\( T^{5} + \)\(42\!\cdots\!27\)\( T^{6} + \)\(12\!\cdots\!88\)\( T^{7} + \)\(37\!\cdots\!35\)\( T^{8} + \)\(95\!\cdots\!62\)\( T^{9} + \)\(25\!\cdots\!70\)\( T^{10} + \)\(65\!\cdots\!68\)\( T^{11} + \)\(25\!\cdots\!70\)\( p^{11} T^{12} + \)\(95\!\cdots\!62\)\( p^{22} T^{13} + \)\(37\!\cdots\!35\)\( p^{33} T^{14} + \)\(12\!\cdots\!88\)\( p^{44} T^{15} + \)\(42\!\cdots\!27\)\( p^{55} T^{16} + \)\(11\!\cdots\!58\)\( p^{66} T^{17} + \)\(29\!\cdots\!62\)\( p^{77} T^{18} + \)\(59\!\cdots\!36\)\( p^{88} T^{19} + \)\(10\!\cdots\!36\)\( p^{99} T^{20} + 120245016462 p^{110} T^{21} + p^{121} T^{22} \)
83 \( 1 + 142463983824 T + \)\(17\!\cdots\!17\)\( T^{2} + \)\(15\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!27\)\( T^{4} + \)\(81\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!51\)\( T^{6} + \)\(25\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} + \)\(55\!\cdots\!28\)\( T^{9} + \)\(22\!\cdots\!66\)\( T^{10} + \)\(84\!\cdots\!92\)\( T^{11} + \)\(22\!\cdots\!66\)\( p^{11} T^{12} + \)\(55\!\cdots\!28\)\( p^{22} T^{13} + \)\(12\!\cdots\!06\)\( p^{33} T^{14} + \)\(25\!\cdots\!72\)\( p^{44} T^{15} + \)\(48\!\cdots\!51\)\( p^{55} T^{16} + \)\(81\!\cdots\!56\)\( p^{66} T^{17} + \)\(12\!\cdots\!27\)\( p^{77} T^{18} + \)\(15\!\cdots\!16\)\( p^{88} T^{19} + \)\(17\!\cdots\!17\)\( p^{99} T^{20} + 142463983824 p^{110} T^{21} + p^{121} T^{22} \)
89 \( 1 + 96700717270 T + \)\(18\!\cdots\!31\)\( T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!39\)\( T^{4} + \)\(95\!\cdots\!74\)\( T^{5} + \)\(87\!\cdots\!01\)\( T^{6} + \)\(47\!\cdots\!92\)\( T^{7} + \)\(37\!\cdots\!86\)\( T^{8} + \)\(18\!\cdots\!32\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} + \)\(55\!\cdots\!32\)\( T^{11} + \)\(12\!\cdots\!62\)\( p^{11} T^{12} + \)\(18\!\cdots\!32\)\( p^{22} T^{13} + \)\(37\!\cdots\!86\)\( p^{33} T^{14} + \)\(47\!\cdots\!92\)\( p^{44} T^{15} + \)\(87\!\cdots\!01\)\( p^{55} T^{16} + \)\(95\!\cdots\!74\)\( p^{66} T^{17} + \)\(15\!\cdots\!39\)\( p^{77} T^{18} + \)\(13\!\cdots\!76\)\( p^{88} T^{19} + \)\(18\!\cdots\!31\)\( p^{99} T^{20} + 96700717270 p^{110} T^{21} + p^{121} T^{22} \)
97 \( 1 + 303190852014 T + \)\(96\!\cdots\!63\)\( T^{2} + \)\(19\!\cdots\!36\)\( p T^{3} + \)\(35\!\cdots\!63\)\( T^{4} + \)\(52\!\cdots\!74\)\( T^{5} + \)\(73\!\cdots\!09\)\( T^{6} + \)\(87\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!14\)\( T^{8} + \)\(10\!\cdots\!80\)\( T^{9} + \)\(98\!\cdots\!18\)\( T^{10} + \)\(84\!\cdots\!72\)\( T^{11} + \)\(98\!\cdots\!18\)\( p^{11} T^{12} + \)\(10\!\cdots\!80\)\( p^{22} T^{13} + \)\(10\!\cdots\!14\)\( p^{33} T^{14} + \)\(87\!\cdots\!00\)\( p^{44} T^{15} + \)\(73\!\cdots\!09\)\( p^{55} T^{16} + \)\(52\!\cdots\!74\)\( p^{66} T^{17} + \)\(35\!\cdots\!63\)\( p^{77} T^{18} + \)\(19\!\cdots\!36\)\( p^{89} T^{19} + \)\(96\!\cdots\!63\)\( p^{99} T^{20} + 303190852014 p^{110} T^{21} + p^{121} T^{22} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.23552079902520007299860888186, −5.17757132729296728175770668576, −5.10064609151974116820436709395, −4.49827824122112019126226150835, −4.40181418560170573681014010080, −4.33908871653396548875867134884, −4.29778887234999796928351970262, −4.03667951182717984168562811124, −3.82667167776660735785974776461, −3.81557952704054681826734558961, −3.77826710354917225734068048621, −3.69745526060338199260234282921, −3.13775074915110377815035613746, −3.03755580094688275101824160969, −2.83652279303155170286532931185, −2.72658368189996727621729630620, −2.60993016213978565941083056424, −2.38301702766475081334257295754, −1.95296979017334912873279818065, −1.92168270465195657911345420568, −1.87420858397928547291201097065, −1.57562838929040543115709607227, −1.24281806890088297027605842609, −1.18598569186602746355060952457, −1.12369909976400620244388924267, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.12369909976400620244388924267, 1.18598569186602746355060952457, 1.24281806890088297027605842609, 1.57562838929040543115709607227, 1.87420858397928547291201097065, 1.92168270465195657911345420568, 1.95296979017334912873279818065, 2.38301702766475081334257295754, 2.60993016213978565941083056424, 2.72658368189996727621729630620, 2.83652279303155170286532931185, 3.03755580094688275101824160969, 3.13775074915110377815035613746, 3.69745526060338199260234282921, 3.77826710354917225734068048621, 3.81557952704054681826734558961, 3.82667167776660735785974776461, 4.03667951182717984168562811124, 4.29778887234999796928351970262, 4.33908871653396548875867134884, 4.40181418560170573681014010080, 4.49827824122112019126226150835, 5.10064609151974116820436709395, 5.17757132729296728175770668576, 5.23552079902520007299860888186

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

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