# 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 of factors

## 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 $$1 + 10999901 T + 423520564906337 T^{2} +$$$$30\!\cdots\!30$$$$T^{3} +$$$$85\!\cdots\!48$$$$T^{4} +$$$$46\!\cdots\!82$$$$T^{5} +$$$$12\!\cdots\!91$$$$T^{6} +$$$$53\!\cdots\!77$$$$T^{7} +$$$$14\!\cdots\!14$$$$T^{8} +$$$$50\!\cdots\!59$$$$T^{9} +$$$$13\!\cdots\!10$$$$T^{10} +$$$$41\!\cdots\!35$$$$T^{11} +$$$$13\!\cdots\!10$$$$p^{7} T^{12} +$$$$50\!\cdots\!59$$$$p^{14} T^{13} +$$$$14\!\cdots\!14$$$$p^{21} T^{14} +$$$$53\!\cdots\!77$$$$p^{28} T^{15} +$$$$12\!\cdots\!91$$$$p^{35} T^{16} +$$$$46\!\cdots\!82$$$$p^{42} T^{17} +$$$$85\!\cdots\!48$$$$p^{49} T^{18} +$$$$30\!\cdots\!30$$$$p^{56} T^{19} + 423520564906337 p^{63} T^{20} + 10999901 p^{70} T^{21} + p^{77} T^{22}$$
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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

−5.61908934275921559223117112473, −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.