Properties

Label 22-43e11-1.1-c7e11-0-0
Degree $22$
Conductor $9.293\times 10^{17}$
Sign $-1$
Analytic cond. $2.56888\times 10^{12}$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $11$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 24·2-s − 68·3-s − 115·4-s − 752·5-s + 1.63e3·6-s − 12·7-s + 7.12e3·8-s − 8.35e3·9-s + 1.80e4·10-s + 1.33e3·11-s + 7.82e3·12-s − 1.79e4·13-s + 288·14-s + 5.11e4·15-s − 4.29e4·16-s − 6.30e4·17-s + 2.00e5·18-s − 5.45e4·19-s + 8.64e4·20-s + 816·21-s − 3.19e4·22-s − 1.38e5·23-s − 4.84e5·24-s − 1.45e5·25-s + 4.31e5·26-s + 6.42e5·27-s + 1.38e3·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.45·3-s − 0.898·4-s − 2.69·5-s + 3.08·6-s − 0.0132·7-s + 4.91·8-s − 3.82·9-s + 5.70·10-s + 0.301·11-s + 1.30·12-s − 2.26·13-s + 0.0280·14-s + 3.91·15-s − 2.62·16-s − 3.11·17-s + 8.10·18-s − 1.82·19-s + 2.41·20-s + 0.0192·21-s − 0.640·22-s − 2.36·23-s − 7.15·24-s − 1.85·25-s + 4.81·26-s + 6.28·27-s + 0.0118·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{11}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(22\)
Conductor: \(43^{11}\)
Sign: $-1$
Analytic conductor: \(2.56888\times 10^{12}\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{43} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(11\)
Selberg data: \((22,\ 43^{11} ,\ ( \ : [7/2]^{11} ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( ( 1 - p^{3} T )^{11} \)
good2 \( 1 + 3 p^{3} T + 691 T^{2} + 6111 p T^{3} + 122431 p T^{4} + 119187 p^{5} T^{5} + 7698897 p^{3} T^{6} + 26761233 p^{5} T^{7} + 372759917 p^{5} T^{8} + 589619691 p^{8} T^{9} + 14564018013 p^{7} T^{10} + 20882571381 p^{10} T^{11} + 14564018013 p^{14} T^{12} + 589619691 p^{22} T^{13} + 372759917 p^{26} T^{14} + 26761233 p^{33} T^{15} + 7698897 p^{38} T^{16} + 119187 p^{47} T^{17} + 122431 p^{50} T^{18} + 6111 p^{57} T^{19} + 691 p^{63} T^{20} + 3 p^{73} T^{21} + p^{77} T^{22} \)
3 \( 1 + 68 T + 12980 T^{2} + 808376 T^{3} + 89267473 T^{4} + 1730409574 p T^{5} + 46563531208 p^{2} T^{6} + 838931847964 p^{3} T^{7} + 18187261065023 p^{4} T^{8} + 301214877851846 p^{5} T^{9} + 5554466143123625 p^{6} T^{10} + 83083651699742984 p^{7} T^{11} + 5554466143123625 p^{13} T^{12} + 301214877851846 p^{19} T^{13} + 18187261065023 p^{25} T^{14} + 838931847964 p^{31} T^{15} + 46563531208 p^{37} T^{16} + 1730409574 p^{43} T^{17} + 89267473 p^{49} T^{18} + 808376 p^{56} T^{19} + 12980 p^{63} T^{20} + 68 p^{70} T^{21} + p^{77} T^{22} \)
5 \( 1 + 752 T + 710712 T^{2} + 398620948 T^{3} + 228483912909 T^{4} + 103902073006684 T^{5} + 45575868805383884 T^{6} + 3502854704787958632 p T^{7} + \)\(25\!\cdots\!43\)\( p^{2} T^{8} + \)\(16\!\cdots\!48\)\( p^{3} T^{9} + \)\(10\!\cdots\!83\)\( p^{4} T^{10} + \)\(60\!\cdots\!84\)\( p^{5} T^{11} + \)\(10\!\cdots\!83\)\( p^{11} T^{12} + \)\(16\!\cdots\!48\)\( p^{17} T^{13} + \)\(25\!\cdots\!43\)\( p^{23} T^{14} + 3502854704787958632 p^{29} T^{15} + 45575868805383884 p^{35} T^{16} + 103902073006684 p^{42} T^{17} + 228483912909 p^{49} T^{18} + 398620948 p^{56} T^{19} + 710712 p^{63} T^{20} + 752 p^{70} T^{21} + p^{77} T^{22} \)
7 \( 1 + 12 T + 3257277 T^{2} - 112854536 T^{3} + 6210749066091 T^{4} - 424770058939316 T^{5} + 8920530970834454311 T^{6} - \)\(59\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(63\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!38\)\( T^{10} - \)\(56\!\cdots\!64\)\( T^{11} + \)\(10\!\cdots\!38\)\( p^{7} T^{12} - \)\(63\!\cdots\!20\)\( p^{14} T^{13} + \)\(10\!\cdots\!10\)\( p^{21} T^{14} - \)\(59\!\cdots\!92\)\( p^{28} T^{15} + 8920530970834454311 p^{35} T^{16} - 424770058939316 p^{42} T^{17} + 6210749066091 p^{49} T^{18} - 112854536 p^{56} T^{19} + 3257277 p^{63} T^{20} + 12 p^{70} T^{21} + p^{77} T^{22} \)
11 \( 1 - 1333 T + 101138498 T^{2} - 129892434295 T^{3} + 5204321338235473 T^{4} - 5619737451849796480 T^{5} + \)\(18\!\cdots\!69\)\( T^{6} - \)\(16\!\cdots\!73\)\( T^{7} + \)\(54\!\cdots\!02\)\( T^{8} - \)\(41\!\cdots\!79\)\( T^{9} + \)\(12\!\cdots\!37\)\( T^{10} - \)\(91\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!37\)\( p^{7} T^{12} - \)\(41\!\cdots\!79\)\( p^{14} T^{13} + \)\(54\!\cdots\!02\)\( p^{21} T^{14} - \)\(16\!\cdots\!73\)\( p^{28} T^{15} + \)\(18\!\cdots\!69\)\( p^{35} T^{16} - 5619737451849796480 p^{42} T^{17} + 5204321338235473 p^{49} T^{18} - 129892434295 p^{56} T^{19} + 101138498 p^{63} T^{20} - 1333 p^{70} T^{21} + p^{77} T^{22} \)
13 \( 1 + 17967 T + 384990808 T^{2} + 3743076892617 T^{3} + 49853715106570123 T^{4} + 27913683973408040352 p T^{5} + \)\(48\!\cdots\!61\)\( T^{6} + \)\(35\!\cdots\!27\)\( T^{7} + \)\(44\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!17\)\( T^{9} + \)\(24\!\cdots\!59\)\( p T^{10} + \)\(18\!\cdots\!72\)\( T^{11} + \)\(24\!\cdots\!59\)\( p^{8} T^{12} + \)\(29\!\cdots\!17\)\( p^{14} T^{13} + \)\(44\!\cdots\!64\)\( p^{21} T^{14} + \)\(35\!\cdots\!27\)\( p^{28} T^{15} + \)\(48\!\cdots\!61\)\( p^{35} T^{16} + 27913683973408040352 p^{43} T^{17} + 49853715106570123 p^{49} T^{18} + 3743076892617 p^{56} T^{19} + 384990808 p^{63} T^{20} + 17967 p^{70} T^{21} + p^{77} T^{22} \)
17 \( 1 + 63095 T + 4595440161 T^{2} + 183012158048722 T^{3} + 7845892593369027108 T^{4} + \)\(23\!\cdots\!66\)\( T^{5} + \)\(75\!\cdots\!23\)\( T^{6} + \)\(18\!\cdots\!63\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} + \)\(10\!\cdots\!53\)\( T^{9} + \)\(25\!\cdots\!74\)\( T^{10} + \)\(48\!\cdots\!85\)\( T^{11} + \)\(25\!\cdots\!74\)\( p^{7} T^{12} + \)\(10\!\cdots\!53\)\( p^{14} T^{13} + \)\(50\!\cdots\!42\)\( p^{21} T^{14} + \)\(18\!\cdots\!63\)\( p^{28} T^{15} + \)\(75\!\cdots\!23\)\( p^{35} T^{16} + \)\(23\!\cdots\!66\)\( p^{42} T^{17} + 7845892593369027108 p^{49} T^{18} + 183012158048722 p^{56} T^{19} + 4595440161 p^{63} T^{20} + 63095 p^{70} T^{21} + p^{77} T^{22} \)
19 \( 1 + 54524 T + 7869332012 T^{2} + 357391219306324 T^{3} + 28295251291043717253 T^{4} + \)\(11\!\cdots\!62\)\( T^{5} + \)\(62\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!64\)\( T^{7} + \)\(98\!\cdots\!19\)\( T^{8} + \)\(30\!\cdots\!82\)\( T^{9} + \)\(11\!\cdots\!53\)\( T^{10} + \)\(31\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!53\)\( p^{7} T^{12} + \)\(30\!\cdots\!82\)\( p^{14} T^{13} + \)\(98\!\cdots\!19\)\( p^{21} T^{14} + \)\(21\!\cdots\!64\)\( p^{28} T^{15} + \)\(62\!\cdots\!16\)\( p^{35} T^{16} + \)\(11\!\cdots\!62\)\( p^{42} T^{17} + 28295251291043717253 p^{49} T^{18} + 357391219306324 p^{56} T^{19} + 7869332012 p^{63} T^{20} + 54524 p^{70} T^{21} + p^{77} T^{22} \)
23 \( 1 + 138139 T + 27512602431 T^{2} + 2853019807837640 T^{3} + \)\(33\!\cdots\!86\)\( T^{4} + \)\(28\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!31\)\( T^{6} + \)\(18\!\cdots\!43\)\( T^{7} + \)\(13\!\cdots\!66\)\( T^{8} + \)\(86\!\cdots\!13\)\( T^{9} + \)\(56\!\cdots\!14\)\( T^{10} + \)\(32\!\cdots\!79\)\( T^{11} + \)\(56\!\cdots\!14\)\( p^{7} T^{12} + \)\(86\!\cdots\!13\)\( p^{14} T^{13} + \)\(13\!\cdots\!66\)\( p^{21} T^{14} + \)\(18\!\cdots\!43\)\( p^{28} T^{15} + \)\(25\!\cdots\!31\)\( p^{35} T^{16} + \)\(28\!\cdots\!30\)\( p^{42} T^{17} + \)\(33\!\cdots\!86\)\( p^{49} T^{18} + 2853019807837640 p^{56} T^{19} + 27512602431 p^{63} T^{20} + 138139 p^{70} T^{21} + p^{77} T^{22} \)
29 \( 1 + 308658 T + 170705721136 T^{2} + 42113858722339938 T^{3} + \)\(13\!\cdots\!57\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!95\)\( T^{8} + \)\(31\!\cdots\!42\)\( T^{9} + \)\(48\!\cdots\!31\)\( T^{10} + \)\(63\!\cdots\!16\)\( T^{11} + \)\(48\!\cdots\!31\)\( p^{7} T^{12} + \)\(31\!\cdots\!42\)\( p^{14} T^{13} + \)\(20\!\cdots\!95\)\( p^{21} T^{14} + \)\(11\!\cdots\!28\)\( p^{28} T^{15} + \)\(63\!\cdots\!20\)\( p^{35} T^{16} + \)\(27\!\cdots\!56\)\( p^{42} T^{17} + \)\(13\!\cdots\!57\)\( p^{49} T^{18} + 42113858722339938 p^{56} T^{19} + 170705721136 p^{63} T^{20} + 308658 p^{70} T^{21} + p^{77} T^{22} \)
31 \( 1 + 209523 T + 225723508099 T^{2} + 41207632211224460 T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(38\!\cdots\!54\)\( T^{5} + \)\(16\!\cdots\!95\)\( T^{6} + \)\(22\!\cdots\!67\)\( T^{7} + \)\(77\!\cdots\!90\)\( T^{8} + \)\(94\!\cdots\!37\)\( T^{9} + \)\(27\!\cdots\!02\)\( T^{10} + \)\(29\!\cdots\!19\)\( T^{11} + \)\(27\!\cdots\!02\)\( p^{7} T^{12} + \)\(94\!\cdots\!37\)\( p^{14} T^{13} + \)\(77\!\cdots\!90\)\( p^{21} T^{14} + \)\(22\!\cdots\!67\)\( p^{28} T^{15} + \)\(16\!\cdots\!95\)\( p^{35} T^{16} + \)\(38\!\cdots\!54\)\( p^{42} T^{17} + \)\(24\!\cdots\!70\)\( p^{49} T^{18} + 41207632211224460 p^{56} T^{19} + 225723508099 p^{63} T^{20} + 209523 p^{70} T^{21} + p^{77} T^{22} \)
37 \( 1 + 298472 T + 366143447012 T^{2} + 1647871098236556 p T^{3} + \)\(20\!\cdots\!85\)\( p T^{4} + \)\(34\!\cdots\!76\)\( p T^{5} + \)\(13\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!72\)\( T^{7} + \)\(17\!\cdots\!31\)\( T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!99\)\( T^{10} + \)\(25\!\cdots\!56\)\( T^{11} + \)\(20\!\cdots\!99\)\( p^{7} T^{12} + \)\(22\!\cdots\!76\)\( p^{14} T^{13} + \)\(17\!\cdots\!31\)\( p^{21} T^{14} + \)\(19\!\cdots\!72\)\( p^{28} T^{15} + \)\(13\!\cdots\!44\)\( p^{35} T^{16} + \)\(34\!\cdots\!76\)\( p^{43} T^{17} + \)\(20\!\cdots\!85\)\( p^{50} T^{18} + 1647871098236556 p^{57} T^{19} + 366143447012 p^{63} T^{20} + 298472 p^{70} T^{21} + p^{77} T^{22} \)
41 \( 1 + 1346735 T + 1406042675633 T^{2} + 1173279952392795098 T^{3} + \)\(86\!\cdots\!00\)\( T^{4} + \)\(57\!\cdots\!74\)\( T^{5} + \)\(35\!\cdots\!19\)\( T^{6} + \)\(19\!\cdots\!47\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(53\!\cdots\!53\)\( T^{9} + \)\(25\!\cdots\!02\)\( T^{10} + \)\(11\!\cdots\!49\)\( T^{11} + \)\(25\!\cdots\!02\)\( p^{7} T^{12} + \)\(53\!\cdots\!53\)\( p^{14} T^{13} + \)\(10\!\cdots\!26\)\( p^{21} T^{14} + \)\(19\!\cdots\!47\)\( p^{28} T^{15} + \)\(35\!\cdots\!19\)\( p^{35} T^{16} + \)\(57\!\cdots\!74\)\( p^{42} T^{17} + \)\(86\!\cdots\!00\)\( p^{49} T^{18} + 1173279952392795098 p^{56} T^{19} + 1406042675633 p^{63} T^{20} + 1346735 p^{70} T^{21} + p^{77} T^{22} \)
47 \( 1 - 499284 T + 4243273022254 T^{2} - 1863388313046890988 T^{3} + \)\(85\!\cdots\!55\)\( T^{4} - \)\(33\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} - \)\(39\!\cdots\!48\)\( T^{7} + \)\(99\!\cdots\!23\)\( T^{8} - \)\(32\!\cdots\!36\)\( T^{9} + \)\(66\!\cdots\!41\)\( T^{10} - \)\(18\!\cdots\!84\)\( T^{11} + \)\(66\!\cdots\!41\)\( p^{7} T^{12} - \)\(32\!\cdots\!36\)\( p^{14} T^{13} + \)\(99\!\cdots\!23\)\( p^{21} T^{14} - \)\(39\!\cdots\!48\)\( p^{28} T^{15} + \)\(11\!\cdots\!96\)\( p^{35} T^{16} - \)\(33\!\cdots\!36\)\( p^{42} T^{17} + \)\(85\!\cdots\!55\)\( p^{49} T^{18} - 1863388313046890988 p^{56} T^{19} + 4243273022254 p^{63} T^{20} - 499284 p^{70} T^{21} + p^{77} T^{22} \)
53 \( 1 + 2210495 T + 10440160443764 T^{2} + 20051339651230402277 T^{3} + \)\(51\!\cdots\!19\)\( T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} + \)\(23\!\cdots\!67\)\( T^{7} + \)\(34\!\cdots\!68\)\( T^{8} + \)\(43\!\cdots\!97\)\( T^{9} + \)\(54\!\cdots\!43\)\( T^{10} + \)\(59\!\cdots\!12\)\( T^{11} + \)\(54\!\cdots\!43\)\( p^{7} T^{12} + \)\(43\!\cdots\!97\)\( p^{14} T^{13} + \)\(34\!\cdots\!68\)\( p^{21} T^{14} + \)\(23\!\cdots\!67\)\( p^{28} T^{15} + \)\(16\!\cdots\!29\)\( p^{35} T^{16} + \)\(86\!\cdots\!24\)\( p^{42} T^{17} + \)\(51\!\cdots\!19\)\( p^{49} T^{18} + 20051339651230402277 p^{56} T^{19} + 10440160443764 p^{63} T^{20} + 2210495 p^{70} T^{21} + p^{77} T^{22} \)
59 \( 1 + 5824216 T + 29093848330233 T^{2} + 94393128322079117696 T^{3} + \)\(28\!\cdots\!95\)\( T^{4} + \)\(68\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!23\)\( T^{6} + \)\(32\!\cdots\!56\)\( T^{7} + \)\(63\!\cdots\!06\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(19\!\cdots\!34\)\( T^{10} + \)\(31\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!34\)\( p^{7} T^{12} + \)\(11\!\cdots\!52\)\( p^{14} T^{13} + \)\(63\!\cdots\!06\)\( p^{21} T^{14} + \)\(32\!\cdots\!56\)\( p^{28} T^{15} + \)\(15\!\cdots\!23\)\( p^{35} T^{16} + \)\(68\!\cdots\!00\)\( p^{42} T^{17} + \)\(28\!\cdots\!95\)\( p^{49} T^{18} + 94393128322079117696 p^{56} T^{19} + 29093848330233 p^{63} T^{20} + 5824216 p^{70} T^{21} + p^{77} T^{22} \)
61 \( 1 + 4453034 T + 27496254230435 T^{2} + 84090456915815697292 T^{3} + \)\(29\!\cdots\!55\)\( T^{4} + \)\(71\!\cdots\!94\)\( T^{5} + \)\(19\!\cdots\!57\)\( T^{6} + \)\(39\!\cdots\!64\)\( T^{7} + \)\(92\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!72\)\( T^{9} + \)\(36\!\cdots\!46\)\( T^{10} + \)\(63\!\cdots\!88\)\( T^{11} + \)\(36\!\cdots\!46\)\( p^{7} T^{12} + \)\(17\!\cdots\!72\)\( p^{14} T^{13} + \)\(92\!\cdots\!06\)\( p^{21} T^{14} + \)\(39\!\cdots\!64\)\( p^{28} T^{15} + \)\(19\!\cdots\!57\)\( p^{35} T^{16} + \)\(71\!\cdots\!94\)\( p^{42} T^{17} + \)\(29\!\cdots\!55\)\( p^{49} T^{18} + 84090456915815697292 p^{56} T^{19} + 27496254230435 p^{63} T^{20} + 4453034 p^{70} T^{21} + p^{77} T^{22} \)
67 \( 1 + 6859513 T + 54299130836534 T^{2} + \)\(22\!\cdots\!67\)\( T^{3} + \)\(10\!\cdots\!85\)\( T^{4} + \)\(33\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} + \)\(37\!\cdots\!61\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!91\)\( T^{9} + \)\(97\!\cdots\!37\)\( T^{10} + \)\(21\!\cdots\!44\)\( T^{11} + \)\(97\!\cdots\!37\)\( p^{7} T^{12} + \)\(32\!\cdots\!91\)\( p^{14} T^{13} + \)\(13\!\cdots\!50\)\( p^{21} T^{14} + \)\(37\!\cdots\!61\)\( p^{28} T^{15} + \)\(13\!\cdots\!77\)\( p^{35} T^{16} + \)\(33\!\cdots\!04\)\( p^{42} T^{17} + \)\(10\!\cdots\!85\)\( p^{49} T^{18} + \)\(22\!\cdots\!67\)\( p^{56} T^{19} + 54299130836534 p^{63} T^{20} + 6859513 p^{70} T^{21} + p^{77} T^{22} \)
71 \( 1 + 10726554 T + 108657874183401 T^{2} + \)\(72\!\cdots\!96\)\( T^{3} + \)\(44\!\cdots\!47\)\( T^{4} + \)\(22\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!71\)\( T^{6} + \)\(43\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!66\)\( T^{8} + \)\(59\!\cdots\!56\)\( T^{9} + \)\(28\!\cdots\!70\)\( p T^{10} + \)\(61\!\cdots\!00\)\( T^{11} + \)\(28\!\cdots\!70\)\( p^{8} T^{12} + \)\(59\!\cdots\!56\)\( p^{14} T^{13} + \)\(16\!\cdots\!66\)\( p^{21} T^{14} + \)\(43\!\cdots\!60\)\( p^{28} T^{15} + \)\(10\!\cdots\!71\)\( p^{35} T^{16} + \)\(22\!\cdots\!78\)\( p^{42} T^{17} + \)\(44\!\cdots\!47\)\( p^{49} T^{18} + \)\(72\!\cdots\!96\)\( p^{56} T^{19} + 108657874183401 p^{63} T^{20} + 10726554 p^{70} T^{21} + p^{77} T^{22} \)
73 \( 1 + 4456898 T + 74251679354831 T^{2} + \)\(30\!\cdots\!24\)\( T^{3} + \)\(37\!\cdots\!83\)\( p T^{4} + \)\(10\!\cdots\!46\)\( T^{5} + \)\(67\!\cdots\!09\)\( T^{6} + \)\(23\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} + \)\(39\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!82\)\( T^{10} + \)\(49\!\cdots\!16\)\( T^{11} + \)\(17\!\cdots\!82\)\( p^{7} T^{12} + \)\(39\!\cdots\!92\)\( p^{14} T^{13} + \)\(12\!\cdots\!46\)\( p^{21} T^{14} + \)\(23\!\cdots\!28\)\( p^{28} T^{15} + \)\(67\!\cdots\!09\)\( p^{35} T^{16} + \)\(10\!\cdots\!46\)\( p^{42} T^{17} + \)\(37\!\cdots\!83\)\( p^{50} T^{18} + \)\(30\!\cdots\!24\)\( p^{56} T^{19} + 74251679354831 p^{63} T^{20} + 4456898 p^{70} T^{21} + p^{77} T^{22} \)
79 \( 1 + 15541320 T + 201904805873602 T^{2} + \)\(17\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!91\)\( T^{4} + \)\(89\!\cdots\!40\)\( T^{5} + \)\(51\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!76\)\( T^{7} + \)\(12\!\cdots\!47\)\( T^{8} + \)\(52\!\cdots\!60\)\( T^{9} + \)\(22\!\cdots\!41\)\( T^{10} + \)\(95\!\cdots\!60\)\( T^{11} + \)\(22\!\cdots\!41\)\( p^{7} T^{12} + \)\(52\!\cdots\!60\)\( p^{14} T^{13} + \)\(12\!\cdots\!47\)\( p^{21} T^{14} + \)\(25\!\cdots\!76\)\( p^{28} T^{15} + \)\(51\!\cdots\!16\)\( p^{35} T^{16} + \)\(89\!\cdots\!40\)\( p^{42} T^{17} + \)\(13\!\cdots\!91\)\( p^{49} T^{18} + \)\(17\!\cdots\!84\)\( p^{56} T^{19} + 201904805873602 p^{63} T^{20} + 15541320 p^{70} T^{21} + p^{77} T^{22} \)
83 \( 1 + 11146767 T + 230549739082006 T^{2} + \)\(20\!\cdots\!69\)\( T^{3} + \)\(24\!\cdots\!77\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!13\)\( T^{6} + \)\(10\!\cdots\!27\)\( T^{7} + \)\(79\!\cdots\!66\)\( T^{8} + \)\(44\!\cdots\!81\)\( T^{9} + \)\(28\!\cdots\!49\)\( T^{10} + \)\(13\!\cdots\!48\)\( T^{11} + \)\(28\!\cdots\!49\)\( p^{7} T^{12} + \)\(44\!\cdots\!81\)\( p^{14} T^{13} + \)\(79\!\cdots\!66\)\( p^{21} T^{14} + \)\(10\!\cdots\!27\)\( p^{28} T^{15} + \)\(16\!\cdots\!13\)\( p^{35} T^{16} + \)\(17\!\cdots\!40\)\( p^{42} T^{17} + \)\(24\!\cdots\!77\)\( p^{49} T^{18} + \)\(20\!\cdots\!69\)\( p^{56} T^{19} + 230549739082006 p^{63} T^{20} + 11146767 p^{70} T^{21} + p^{77} T^{22} \)
89 \( 1 + 13531356 T + 311067335905063 T^{2} + \)\(31\!\cdots\!84\)\( T^{3} + \)\(45\!\cdots\!03\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(45\!\cdots\!69\)\( T^{6} + \)\(34\!\cdots\!88\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} + \)\(21\!\cdots\!92\)\( T^{9} + \)\(18\!\cdots\!18\)\( T^{10} + \)\(10\!\cdots\!68\)\( T^{11} + \)\(18\!\cdots\!18\)\( p^{7} T^{12} + \)\(21\!\cdots\!92\)\( p^{14} T^{13} + \)\(32\!\cdots\!22\)\( p^{21} T^{14} + \)\(34\!\cdots\!88\)\( p^{28} T^{15} + \)\(45\!\cdots\!69\)\( p^{35} T^{16} + \)\(39\!\cdots\!56\)\( p^{42} T^{17} + \)\(45\!\cdots\!03\)\( p^{49} T^{18} + \)\(31\!\cdots\!84\)\( p^{56} T^{19} + 311067335905063 p^{63} T^{20} + 13531356 p^{70} T^{21} + p^{77} T^{22} \)
97 \( 1 + 10999901 T + 423520564906337 T^{2} + \)\(30\!\cdots\!30\)\( T^{3} + \)\(85\!\cdots\!48\)\( T^{4} + \)\(46\!\cdots\!82\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} + \)\(53\!\cdots\!77\)\( T^{7} + \)\(14\!\cdots\!14\)\( T^{8} + \)\(50\!\cdots\!59\)\( T^{9} + \)\(13\!\cdots\!10\)\( T^{10} + \)\(41\!\cdots\!35\)\( T^{11} + \)\(13\!\cdots\!10\)\( p^{7} T^{12} + \)\(50\!\cdots\!59\)\( p^{14} T^{13} + \)\(14\!\cdots\!14\)\( p^{21} T^{14} + \)\(53\!\cdots\!77\)\( p^{28} T^{15} + \)\(12\!\cdots\!91\)\( p^{35} T^{16} + \)\(46\!\cdots\!82\)\( p^{42} T^{17} + \)\(85\!\cdots\!48\)\( p^{49} T^{18} + \)\(30\!\cdots\!30\)\( p^{56} T^{19} + 423520564906337 p^{63} T^{20} + 10999901 p^{70} T^{21} + p^{77} T^{22} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.61908934275921559223117112473, −5.42933252132158105279171966710, −5.23970219426776687566954600673, −5.02474729284303870685524961306, −4.71022872187411868986256535883, −4.65801383280346497820947174918, −4.51014645997480542315599378891, −4.38407571352645288159798361556, −4.22252855248802122035494951355, −4.22003505495745856128667345440, −4.21740270220147840353974682427, −3.77114134812089103431181782234, −3.66110454699516717811952264450, −3.55933564106661000678463628052, −3.09963917666605238842695124179, −3.08344799165299122571213337901, −2.96438791237467688028337056439, −2.59540400657507680730583755510, −2.56332852367118828090026655227, −2.08999453631188021821557746347, −1.96459362656104226334755752141, −1.84616081283351712840215811817, −1.69707856900039546666902939838, −1.35380385121915730978728954136, −1.26161444862495668862143372586, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.26161444862495668862143372586, 1.35380385121915730978728954136, 1.69707856900039546666902939838, 1.84616081283351712840215811817, 1.96459362656104226334755752141, 2.08999453631188021821557746347, 2.56332852367118828090026655227, 2.59540400657507680730583755510, 2.96438791237467688028337056439, 3.08344799165299122571213337901, 3.09963917666605238842695124179, 3.55933564106661000678463628052, 3.66110454699516717811952264450, 3.77114134812089103431181782234, 4.21740270220147840353974682427, 4.22003505495745856128667345440, 4.22252855248802122035494951355, 4.38407571352645288159798361556, 4.51014645997480542315599378891, 4.65801383280346497820947174918, 4.71022872187411868986256535883, 5.02474729284303870685524961306, 5.23970219426776687566954600673, 5.42933252132158105279171966710, 5.61908934275921559223117112473

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

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