# Properties

 Degree 30 Conductor $43^{15}$ Sign $-1$ Motivic weight 9 Primitive no Self-dual yes Analytic rank 15

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·2-s − 317·3-s − 1.70e3·4-s − 4.71e3·5-s + 1.01e4·6-s − 9.68e3·7-s + 6.42e4·8-s − 6.26e4·9-s + 1.50e5·10-s − 1.04e5·11-s + 5.41e5·12-s − 1.16e5·13-s + 3.09e5·14-s + 1.49e6·15-s + 1.24e6·16-s − 8.84e5·17-s + 2.00e6·18-s − 6.89e5·19-s + 8.05e6·20-s + 3.06e6·21-s + 3.34e6·22-s − 2.50e6·23-s − 2.03e7·24-s − 2.86e6·25-s + 3.71e6·26-s + 2.83e7·27-s + 1.65e7·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.25·3-s − 3.33·4-s − 3.37·5-s + 3.19·6-s − 1.52·7-s + 5.54·8-s − 3.18·9-s + 4.77·10-s − 2.15·11-s + 7.53·12-s − 1.12·13-s + 2.15·14-s + 7.62·15-s + 4.75·16-s − 2.56·17-s + 4.49·18-s − 1.21·19-s + 11.2·20-s + 3.44·21-s + 3.04·22-s − 1.86·23-s − 12.5·24-s − 1.46·25-s + 1.59·26-s + 10.2·27-s + 5.08·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{15}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{15} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$30$$ $$N$$ = $$43^{15}$$ $$\varepsilon$$ = $-1$ motivic weight = $$9$$ character : induced by $\chi_{43} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$15$$ Selberg data = $$(30,\ 43^{15} ,\ ( \ : [9/2]^{15} ),\ -1 )$$ $$L(5)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 30. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 29.
$p$$F_p(T)$
bad43 $$( 1 + p^{4} T )^{15}$$
good2 $$1 + p^{5} T + 2731 T^{2} + 38903 p T^{3} + 1925179 p T^{4} + 12729931 p^{3} T^{5} + 466748515 p^{3} T^{6} + 2790127665 p^{5} T^{7} + 42541932853 p^{6} T^{8} + 226560437741 p^{8} T^{9} + 3119560601683 p^{9} T^{10} + 14760976902969 p^{11} T^{11} + 48909112098911 p^{14} T^{12} + 213220815288751 p^{16} T^{13} + 2927267306494753 p^{17} T^{14} + 12819018469580755 p^{19} T^{15} + 2927267306494753 p^{26} T^{16} + 213220815288751 p^{34} T^{17} + 48909112098911 p^{41} T^{18} + 14760976902969 p^{47} T^{19} + 3119560601683 p^{54} T^{20} + 226560437741 p^{62} T^{21} + 42541932853 p^{69} T^{22} + 2790127665 p^{77} T^{23} + 466748515 p^{84} T^{24} + 12729931 p^{93} T^{25} + 1925179 p^{100} T^{26} + 38903 p^{109} T^{27} + 2731 p^{117} T^{28} + p^{131} T^{29} + p^{135} T^{30}$$
3 $$1 + 317 T + 163109 T^{2} + 14396569 p T^{3} + 13657333057 T^{4} + 3049240516432 T^{5} + 749507085253976 T^{6} + 1815939834537548 p^{4} T^{7} + 1130216186040245534 p^{3} T^{8} +$$$$19\!\cdots\!38$$$$p^{3} T^{9} +$$$$40\!\cdots\!23$$$$p^{5} T^{10} +$$$$21\!\cdots\!09$$$$p^{6} T^{11} +$$$$36\!\cdots\!85$$$$p^{6} T^{12} +$$$$60\!\cdots\!61$$$$p^{8} T^{13} +$$$$30\!\cdots\!53$$$$p^{9} T^{14} +$$$$42\!\cdots\!40$$$$p^{9} T^{15} +$$$$30\!\cdots\!53$$$$p^{18} T^{16} +$$$$60\!\cdots\!61$$$$p^{26} T^{17} +$$$$36\!\cdots\!85$$$$p^{33} T^{18} +$$$$21\!\cdots\!09$$$$p^{42} T^{19} +$$$$40\!\cdots\!23$$$$p^{50} T^{20} +$$$$19\!\cdots\!38$$$$p^{57} T^{21} + 1130216186040245534 p^{66} T^{22} + 1815939834537548 p^{76} T^{23} + 749507085253976 p^{81} T^{24} + 3049240516432 p^{90} T^{25} + 13657333057 p^{99} T^{26} + 14396569 p^{109} T^{27} + 163109 p^{117} T^{28} + 317 p^{126} T^{29} + p^{135} T^{30}$$
5 $$1 + 4717 T + 25115807 T^{2} + 83707229397 T^{3} + 275195357593257 T^{4} + 146051295003353118 p T^{5} +$$$$18\!\cdots\!78$$$$T^{6} +$$$$41\!\cdots\!18$$$$T^{7} +$$$$89\!\cdots\!28$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$32\!\cdots\!59$$$$T^{10} +$$$$11\!\cdots\!91$$$$p T^{11} +$$$$38\!\cdots\!17$$$$p^{2} T^{12} +$$$$12\!\cdots\!89$$$$p^{3} T^{13} +$$$$36\!\cdots\!97$$$$p^{4} T^{14} +$$$$10\!\cdots\!08$$$$p^{5} T^{15} +$$$$36\!\cdots\!97$$$$p^{13} T^{16} +$$$$12\!\cdots\!89$$$$p^{21} T^{17} +$$$$38\!\cdots\!17$$$$p^{29} T^{18} +$$$$11\!\cdots\!91$$$$p^{37} T^{19} +$$$$32\!\cdots\!59$$$$p^{45} T^{20} +$$$$17\!\cdots\!68$$$$p^{54} T^{21} +$$$$89\!\cdots\!28$$$$p^{63} T^{22} +$$$$41\!\cdots\!18$$$$p^{72} T^{23} +$$$$18\!\cdots\!78$$$$p^{81} T^{24} + 146051295003353118 p^{91} T^{25} + 275195357593257 p^{99} T^{26} + 83707229397 p^{108} T^{27} + 25115807 p^{117} T^{28} + 4717 p^{126} T^{29} + p^{135} T^{30}$$
7 $$1 + 9680 T + 303171705 T^{2} + 2420677168232 T^{3} + 5969978681963587 p T^{4} +$$$$27\!\cdots\!60$$$$T^{5} +$$$$34\!\cdots\!17$$$$T^{6} +$$$$25\!\cdots\!12$$$$p T^{7} +$$$$37\!\cdots\!89$$$$p^{2} T^{8} +$$$$18\!\cdots\!52$$$$p^{3} T^{9} +$$$$24\!\cdots\!33$$$$p^{4} T^{10} -$$$$69\!\cdots\!88$$$$p^{6} T^{11} +$$$$45\!\cdots\!97$$$$p^{6} T^{12} -$$$$17\!\cdots\!52$$$$p^{7} T^{13} -$$$$76\!\cdots\!71$$$$p^{8} T^{14} -$$$$29\!\cdots\!84$$$$p^{10} T^{15} -$$$$76\!\cdots\!71$$$$p^{17} T^{16} -$$$$17\!\cdots\!52$$$$p^{25} T^{17} +$$$$45\!\cdots\!97$$$$p^{33} T^{18} -$$$$69\!\cdots\!88$$$$p^{42} T^{19} +$$$$24\!\cdots\!33$$$$p^{49} T^{20} +$$$$18\!\cdots\!52$$$$p^{57} T^{21} +$$$$37\!\cdots\!89$$$$p^{65} T^{22} +$$$$25\!\cdots\!12$$$$p^{73} T^{23} +$$$$34\!\cdots\!17$$$$p^{81} T^{24} +$$$$27\!\cdots\!60$$$$p^{90} T^{25} + 5969978681963587 p^{100} T^{26} + 2420677168232 p^{108} T^{27} + 303171705 p^{117} T^{28} + 9680 p^{126} T^{29} + p^{135} T^{30}$$
11 $$1 + 104484 T + 21671047097 T^{2} + 1826154580412208 T^{3} +$$$$22\!\cdots\!79$$$$T^{4} +$$$$16\!\cdots\!04$$$$T^{5} +$$$$15\!\cdots\!71$$$$T^{6} +$$$$10\!\cdots\!88$$$$T^{7} +$$$$81\!\cdots\!64$$$$T^{8} +$$$$46\!\cdots\!00$$$$T^{9} +$$$$29\!\cdots\!64$$$$p T^{10} +$$$$17\!\cdots\!28$$$$T^{11} +$$$$10\!\cdots\!38$$$$T^{12} +$$$$52\!\cdots\!68$$$$T^{13} +$$$$27\!\cdots\!22$$$$p T^{14} +$$$$13\!\cdots\!92$$$$T^{15} +$$$$27\!\cdots\!22$$$$p^{10} T^{16} +$$$$52\!\cdots\!68$$$$p^{18} T^{17} +$$$$10\!\cdots\!38$$$$p^{27} T^{18} +$$$$17\!\cdots\!28$$$$p^{36} T^{19} +$$$$29\!\cdots\!64$$$$p^{46} T^{20} +$$$$46\!\cdots\!00$$$$p^{54} T^{21} +$$$$81\!\cdots\!64$$$$p^{63} T^{22} +$$$$10\!\cdots\!88$$$$p^{72} T^{23} +$$$$15\!\cdots\!71$$$$p^{81} T^{24} +$$$$16\!\cdots\!04$$$$p^{90} T^{25} +$$$$22\!\cdots\!79$$$$p^{99} T^{26} + 1826154580412208 p^{108} T^{27} + 21671047097 p^{117} T^{28} + 104484 p^{126} T^{29} + p^{135} T^{30}$$
13 $$1 + 116174 T + 93110646219 T^{2} + 9939551633003756 T^{3} +$$$$33\!\cdots\!91$$$$p T^{4} +$$$$42\!\cdots\!54$$$$T^{5} +$$$$13\!\cdots\!41$$$$T^{6} +$$$$12\!\cdots\!40$$$$T^{7} +$$$$31\!\cdots\!88$$$$T^{8} +$$$$25\!\cdots\!00$$$$T^{9} +$$$$57\!\cdots\!96$$$$T^{10} +$$$$42\!\cdots\!04$$$$T^{11} +$$$$85\!\cdots\!78$$$$T^{12} +$$$$58\!\cdots\!52$$$$T^{13} +$$$$10\!\cdots\!26$$$$T^{14} +$$$$67\!\cdots\!12$$$$T^{15} +$$$$10\!\cdots\!26$$$$p^{9} T^{16} +$$$$58\!\cdots\!52$$$$p^{18} T^{17} +$$$$85\!\cdots\!78$$$$p^{27} T^{18} +$$$$42\!\cdots\!04$$$$p^{36} T^{19} +$$$$57\!\cdots\!96$$$$p^{45} T^{20} +$$$$25\!\cdots\!00$$$$p^{54} T^{21} +$$$$31\!\cdots\!88$$$$p^{63} T^{22} +$$$$12\!\cdots\!40$$$$p^{72} T^{23} +$$$$13\!\cdots\!41$$$$p^{81} T^{24} +$$$$42\!\cdots\!54$$$$p^{90} T^{25} +$$$$33\!\cdots\!91$$$$p^{100} T^{26} + 9939551633003756 p^{108} T^{27} + 93110646219 p^{117} T^{28} + 116174 p^{126} T^{29} + p^{135} T^{30}$$
17 $$1 + 884265 T + 1384046763853 T^{2} + 1008991515705553461 T^{3} +$$$$92\!\cdots\!03$$$$T^{4} +$$$$56\!\cdots\!24$$$$T^{5} +$$$$38\!\cdots\!52$$$$T^{6} +$$$$20\!\cdots\!80$$$$T^{7} +$$$$11\!\cdots\!15$$$$T^{8} +$$$$54\!\cdots\!13$$$$T^{9} +$$$$26\!\cdots\!40$$$$T^{10} +$$$$11\!\cdots\!12$$$$T^{11} +$$$$47\!\cdots\!38$$$$T^{12} +$$$$17\!\cdots\!77$$$$T^{13} +$$$$69\!\cdots\!95$$$$T^{14} +$$$$23\!\cdots\!62$$$$T^{15} +$$$$69\!\cdots\!95$$$$p^{9} T^{16} +$$$$17\!\cdots\!77$$$$p^{18} T^{17} +$$$$47\!\cdots\!38$$$$p^{27} T^{18} +$$$$11\!\cdots\!12$$$$p^{36} T^{19} +$$$$26\!\cdots\!40$$$$p^{45} T^{20} +$$$$54\!\cdots\!13$$$$p^{54} T^{21} +$$$$11\!\cdots\!15$$$$p^{63} T^{22} +$$$$20\!\cdots\!80$$$$p^{72} T^{23} +$$$$38\!\cdots\!52$$$$p^{81} T^{24} +$$$$56\!\cdots\!24$$$$p^{90} T^{25} +$$$$92\!\cdots\!03$$$$p^{99} T^{26} + 1008991515705553461 p^{108} T^{27} + 1384046763853 p^{117} T^{28} + 884265 p^{126} T^{29} + p^{135} T^{30}$$
19 $$1 + 689535 T + 2651913177639 T^{2} + 1907898195520184901 T^{3} +$$$$36\!\cdots\!69$$$$T^{4} +$$$$26\!\cdots\!68$$$$T^{5} +$$$$34\!\cdots\!48$$$$T^{6} +$$$$13\!\cdots\!96$$$$p T^{7} +$$$$24\!\cdots\!78$$$$T^{8} +$$$$17\!\cdots\!10$$$$T^{9} +$$$$14\!\cdots\!57$$$$T^{10} +$$$$48\!\cdots\!09$$$$p T^{11} +$$$$66\!\cdots\!35$$$$T^{12} +$$$$40\!\cdots\!15$$$$T^{13} +$$$$25\!\cdots\!19$$$$T^{14} +$$$$14\!\cdots\!88$$$$T^{15} +$$$$25\!\cdots\!19$$$$p^{9} T^{16} +$$$$40\!\cdots\!15$$$$p^{18} T^{17} +$$$$66\!\cdots\!35$$$$p^{27} T^{18} +$$$$48\!\cdots\!09$$$$p^{37} T^{19} +$$$$14\!\cdots\!57$$$$p^{45} T^{20} +$$$$17\!\cdots\!10$$$$p^{54} T^{21} +$$$$24\!\cdots\!78$$$$p^{63} T^{22} +$$$$13\!\cdots\!96$$$$p^{73} T^{23} +$$$$34\!\cdots\!48$$$$p^{81} T^{24} +$$$$26\!\cdots\!68$$$$p^{90} T^{25} +$$$$36\!\cdots\!69$$$$p^{99} T^{26} + 1907898195520184901 p^{108} T^{27} + 2651913177639 p^{117} T^{28} + 689535 p^{126} T^{29} + p^{135} T^{30}$$
23 $$1 + 2504077 T + 862494392727 p T^{2} + 44904598844137902367 T^{3} +$$$$19\!\cdots\!31$$$$T^{4} +$$$$39\!\cdots\!74$$$$T^{5} +$$$$12\!\cdots\!34$$$$T^{6} +$$$$22\!\cdots\!78$$$$T^{7} +$$$$54\!\cdots\!85$$$$T^{8} +$$$$92\!\cdots\!11$$$$T^{9} +$$$$19\!\cdots\!50$$$$T^{10} +$$$$29\!\cdots\!96$$$$T^{11} +$$$$52\!\cdots\!24$$$$T^{12} +$$$$73\!\cdots\!07$$$$T^{13} +$$$$11\!\cdots\!35$$$$T^{14} +$$$$14\!\cdots\!00$$$$T^{15} +$$$$11\!\cdots\!35$$$$p^{9} T^{16} +$$$$73\!\cdots\!07$$$$p^{18} T^{17} +$$$$52\!\cdots\!24$$$$p^{27} T^{18} +$$$$29\!\cdots\!96$$$$p^{36} T^{19} +$$$$19\!\cdots\!50$$$$p^{45} T^{20} +$$$$92\!\cdots\!11$$$$p^{54} T^{21} +$$$$54\!\cdots\!85$$$$p^{63} T^{22} +$$$$22\!\cdots\!78$$$$p^{72} T^{23} +$$$$12\!\cdots\!34$$$$p^{81} T^{24} +$$$$39\!\cdots\!74$$$$p^{90} T^{25} +$$$$19\!\cdots\!31$$$$p^{99} T^{26} + 44904598844137902367 p^{108} T^{27} + 862494392727 p^{118} T^{28} + 2504077 p^{126} T^{29} + p^{135} T^{30}$$
29 $$1 + 18406221 T + 231322256139583 T^{2} +$$$$21\!\cdots\!69$$$$T^{3} +$$$$16\!\cdots\!45$$$$T^{4} +$$$$10\!\cdots\!86$$$$T^{5} +$$$$64\!\cdots\!34$$$$T^{6} +$$$$34\!\cdots\!58$$$$T^{7} +$$$$16\!\cdots\!04$$$$T^{8} +$$$$73\!\cdots\!72$$$$T^{9} +$$$$30\!\cdots\!31$$$$T^{10} +$$$$11\!\cdots\!11$$$$T^{11} +$$$$44\!\cdots\!21$$$$T^{12} +$$$$16\!\cdots\!53$$$$T^{13} +$$$$60\!\cdots\!37$$$$T^{14} +$$$$22\!\cdots\!12$$$$T^{15} +$$$$60\!\cdots\!37$$$$p^{9} T^{16} +$$$$16\!\cdots\!53$$$$p^{18} T^{17} +$$$$44\!\cdots\!21$$$$p^{27} T^{18} +$$$$11\!\cdots\!11$$$$p^{36} T^{19} +$$$$30\!\cdots\!31$$$$p^{45} T^{20} +$$$$73\!\cdots\!72$$$$p^{54} T^{21} +$$$$16\!\cdots\!04$$$$p^{63} T^{22} +$$$$34\!\cdots\!58$$$$p^{72} T^{23} +$$$$64\!\cdots\!34$$$$p^{81} T^{24} +$$$$10\!\cdots\!86$$$$p^{90} T^{25} +$$$$16\!\cdots\!45$$$$p^{99} T^{26} +$$$$21\!\cdots\!69$$$$p^{108} T^{27} + 231322256139583 p^{117} T^{28} + 18406221 p^{126} T^{29} + p^{135} T^{30}$$
31 $$1 + 12033699 T + 157221027222111 T^{2} +$$$$11\!\cdots\!53$$$$T^{3} +$$$$11\!\cdots\!51$$$$T^{4} +$$$$78\!\cdots\!70$$$$T^{5} +$$$$65\!\cdots\!70$$$$T^{6} +$$$$42\!\cdots\!02$$$$T^{7} +$$$$29\!\cdots\!41$$$$T^{8} +$$$$18\!\cdots\!53$$$$T^{9} +$$$$11\!\cdots\!80$$$$T^{10} +$$$$68\!\cdots\!12$$$$T^{11} +$$$$39\!\cdots\!74$$$$T^{12} +$$$$21\!\cdots\!69$$$$T^{13} +$$$$11\!\cdots\!43$$$$T^{14} +$$$$62\!\cdots\!76$$$$T^{15} +$$$$11\!\cdots\!43$$$$p^{9} T^{16} +$$$$21\!\cdots\!69$$$$p^{18} T^{17} +$$$$39\!\cdots\!74$$$$p^{27} T^{18} +$$$$68\!\cdots\!12$$$$p^{36} T^{19} +$$$$11\!\cdots\!80$$$$p^{45} T^{20} +$$$$18\!\cdots\!53$$$$p^{54} T^{21} +$$$$29\!\cdots\!41$$$$p^{63} T^{22} +$$$$42\!\cdots\!02$$$$p^{72} T^{23} +$$$$65\!\cdots\!70$$$$p^{81} T^{24} +$$$$78\!\cdots\!70$$$$p^{90} T^{25} +$$$$11\!\cdots\!51$$$$p^{99} T^{26} +$$$$11\!\cdots\!53$$$$p^{108} T^{27} + 157221027222111 p^{117} T^{28} + 12033699 p^{126} T^{29} + p^{135} T^{30}$$
37 $$1 + 8722847 T + 568554278402447 T^{2} +$$$$60\!\cdots\!91$$$$T^{3} +$$$$19\!\cdots\!13$$$$T^{4} +$$$$18\!\cdots\!10$$$$T^{5} +$$$$47\!\cdots\!42$$$$T^{6} +$$$$39\!\cdots\!30$$$$T^{7} +$$$$80\!\cdots\!88$$$$T^{8} +$$$$52\!\cdots\!84$$$$T^{9} +$$$$99\!\cdots\!43$$$$T^{10} +$$$$39\!\cdots\!97$$$$T^{11} +$$$$88\!\cdots\!13$$$$T^{12} -$$$$31\!\cdots\!73$$$$T^{13} +$$$$63\!\cdots\!65$$$$T^{14} -$$$$36\!\cdots\!60$$$$T^{15} +$$$$63\!\cdots\!65$$$$p^{9} T^{16} -$$$$31\!\cdots\!73$$$$p^{18} T^{17} +$$$$88\!\cdots\!13$$$$p^{27} T^{18} +$$$$39\!\cdots\!97$$$$p^{36} T^{19} +$$$$99\!\cdots\!43$$$$p^{45} T^{20} +$$$$52\!\cdots\!84$$$$p^{54} T^{21} +$$$$80\!\cdots\!88$$$$p^{63} T^{22} +$$$$39\!\cdots\!30$$$$p^{72} T^{23} +$$$$47\!\cdots\!42$$$$p^{81} T^{24} +$$$$18\!\cdots\!10$$$$p^{90} T^{25} +$$$$19\!\cdots\!13$$$$p^{99} T^{26} +$$$$60\!\cdots\!91$$$$p^{108} T^{27} + 568554278402447 p^{117} T^{28} + 8722847 p^{126} T^{29} + p^{135} T^{30}$$
41 $$1 + 18689389 T + 2776615386417537 T^{2} +$$$$46\!\cdots\!33$$$$T^{3} +$$$$91\!\cdots\!39$$$$p T^{4} +$$$$58\!\cdots\!96$$$$T^{5} +$$$$33\!\cdots\!84$$$$T^{6} +$$$$48\!\cdots\!36$$$$T^{7} +$$$$21\!\cdots\!31$$$$T^{8} +$$$$30\!\cdots\!37$$$$T^{9} +$$$$11\!\cdots\!84$$$$T^{10} +$$$$15\!\cdots\!00$$$$T^{11} +$$$$48\!\cdots\!18$$$$T^{12} +$$$$63\!\cdots\!65$$$$T^{13} +$$$$18\!\cdots\!43$$$$T^{14} +$$$$22\!\cdots\!26$$$$T^{15} +$$$$18\!\cdots\!43$$$$p^{9} T^{16} +$$$$63\!\cdots\!65$$$$p^{18} T^{17} +$$$$48\!\cdots\!18$$$$p^{27} T^{18} +$$$$15\!\cdots\!00$$$$p^{36} T^{19} +$$$$11\!\cdots\!84$$$$p^{45} T^{20} +$$$$30\!\cdots\!37$$$$p^{54} T^{21} +$$$$21\!\cdots\!31$$$$p^{63} T^{22} +$$$$48\!\cdots\!36$$$$p^{72} T^{23} +$$$$33\!\cdots\!84$$$$p^{81} T^{24} +$$$$58\!\cdots\!96$$$$p^{90} T^{25} +$$$$91\!\cdots\!39$$$$p^{100} T^{26} +$$$$46\!\cdots\!33$$$$p^{108} T^{27} + 2776615386417537 p^{117} T^{28} + 18689389 p^{126} T^{29} + p^{135} T^{30}$$
47 $$1 + 104960741 T + 13310159754260471 T^{2} +$$$$95\!\cdots\!91$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} +$$$$92\!\cdots\!28$$$$p T^{5} +$$$$26\!\cdots\!08$$$$T^{6} +$$$$13\!\cdots\!76$$$$T^{7} +$$$$66\!\cdots\!90$$$$T^{8} +$$$$28\!\cdots\!38$$$$T^{9} +$$$$12\!\cdots\!89$$$$T^{10} +$$$$49\!\cdots\!97$$$$T^{11} +$$$$20\!\cdots\!91$$$$T^{12} +$$$$70\!\cdots\!13$$$$T^{13} +$$$$26\!\cdots\!11$$$$T^{14} +$$$$85\!\cdots\!24$$$$T^{15} +$$$$26\!\cdots\!11$$$$p^{9} T^{16} +$$$$70\!\cdots\!13$$$$p^{18} T^{17} +$$$$20\!\cdots\!91$$$$p^{27} T^{18} +$$$$49\!\cdots\!97$$$$p^{36} T^{19} +$$$$12\!\cdots\!89$$$$p^{45} T^{20} +$$$$28\!\cdots\!38$$$$p^{54} T^{21} +$$$$66\!\cdots\!90$$$$p^{63} T^{22} +$$$$13\!\cdots\!76$$$$p^{72} T^{23} +$$$$26\!\cdots\!08$$$$p^{81} T^{24} +$$$$92\!\cdots\!28$$$$p^{91} T^{25} +$$$$75\!\cdots\!21$$$$p^{99} T^{26} +$$$$95\!\cdots\!91$$$$p^{108} T^{27} + 13310159754260471 p^{117} T^{28} + 104960741 p^{126} T^{29} + p^{135} T^{30}$$
53 $$1 + 215907800 T + 45601589020862355 T^{2} +$$$$63\!\cdots\!44$$$$T^{3} +$$$$82\!\cdots\!07$$$$T^{4} +$$$$85\!\cdots\!20$$$$T^{5} +$$$$82\!\cdots\!01$$$$T^{6} +$$$$67\!\cdots\!72$$$$T^{7} +$$$$51\!\cdots\!60$$$$T^{8} +$$$$34\!\cdots\!76$$$$T^{9} +$$$$21\!\cdots\!24$$$$T^{10} +$$$$11\!\cdots\!24$$$$T^{11} +$$$$58\!\cdots\!90$$$$T^{12} +$$$$26\!\cdots\!88$$$$T^{13} +$$$$12\!\cdots\!62$$$$T^{14} +$$$$61\!\cdots\!84$$$$T^{15} +$$$$12\!\cdots\!62$$$$p^{9} T^{16} +$$$$26\!\cdots\!88$$$$p^{18} T^{17} +$$$$58\!\cdots\!90$$$$p^{27} T^{18} +$$$$11\!\cdots\!24$$$$p^{36} T^{19} +$$$$21\!\cdots\!24$$$$p^{45} T^{20} +$$$$34\!\cdots\!76$$$$p^{54} T^{21} +$$$$51\!\cdots\!60$$$$p^{63} T^{22} +$$$$67\!\cdots\!72$$$$p^{72} T^{23} +$$$$82\!\cdots\!01$$$$p^{81} T^{24} +$$$$85\!\cdots\!20$$$$p^{90} T^{25} +$$$$82\!\cdots\!07$$$$p^{99} T^{26} +$$$$63\!\cdots\!44$$$$p^{108} T^{27} + 45601589020862355 p^{117} T^{28} + 215907800 p^{126} T^{29} + p^{135} T^{30}$$
59 $$1 - 185924544 T + 72641643327616985 T^{2} -$$$$11\!\cdots\!56$$$$T^{3} +$$$$26\!\cdots\!25$$$$T^{4} -$$$$37\!\cdots\!76$$$$T^{5} +$$$$67\!\cdots\!65$$$$T^{6} -$$$$85\!\cdots\!64$$$$T^{7} +$$$$12\!\cdots\!49$$$$T^{8} -$$$$14\!\cdots\!84$$$$T^{9} +$$$$19\!\cdots\!45$$$$T^{10} -$$$$20\!\cdots\!16$$$$T^{11} +$$$$24\!\cdots\!49$$$$T^{12} -$$$$23\!\cdots\!92$$$$T^{13} +$$$$25\!\cdots\!69$$$$T^{14} -$$$$22\!\cdots\!84$$$$T^{15} +$$$$25\!\cdots\!69$$$$p^{9} T^{16} -$$$$23\!\cdots\!92$$$$p^{18} T^{17} +$$$$24\!\cdots\!49$$$$p^{27} T^{18} -$$$$20\!\cdots\!16$$$$p^{36} T^{19} +$$$$19\!\cdots\!45$$$$p^{45} T^{20} -$$$$14\!\cdots\!84$$$$p^{54} T^{21} +$$$$12\!\cdots\!49$$$$p^{63} T^{22} -$$$$85\!\cdots\!64$$$$p^{72} T^{23} +$$$$67\!\cdots\!65$$$$p^{81} T^{24} -$$$$37\!\cdots\!76$$$$p^{90} T^{25} +$$$$26\!\cdots\!25$$$$p^{99} T^{26} -$$$$11\!\cdots\!56$$$$p^{108} T^{27} + 72641643327616985 p^{117} T^{28} - 185924544 p^{126} T^{29} + p^{135} T^{30}$$
61 $$1 - 247538102 T + 123939924002267915 T^{2} -$$$$25\!\cdots\!76$$$$T^{3} +$$$$72\!\cdots\!53$$$$T^{4} -$$$$12\!\cdots\!30$$$$T^{5} +$$$$27\!\cdots\!23$$$$T^{6} -$$$$42\!\cdots\!04$$$$T^{7} +$$$$73\!\cdots\!05$$$$T^{8} -$$$$10\!\cdots\!70$$$$T^{9} +$$$$15\!\cdots\!63$$$$T^{10} -$$$$19\!\cdots\!64$$$$T^{11} +$$$$26\!\cdots\!33$$$$T^{12} -$$$$30\!\cdots\!38$$$$T^{13} +$$$$37\!\cdots\!95$$$$T^{14} -$$$$39\!\cdots\!08$$$$T^{15} +$$$$37\!\cdots\!95$$$$p^{9} T^{16} -$$$$30\!\cdots\!38$$$$p^{18} T^{17} +$$$$26\!\cdots\!33$$$$p^{27} T^{18} -$$$$19\!\cdots\!64$$$$p^{36} T^{19} +$$$$15\!\cdots\!63$$$$p^{45} T^{20} -$$$$10\!\cdots\!70$$$$p^{54} T^{21} +$$$$73\!\cdots\!05$$$$p^{63} T^{22} -$$$$42\!\cdots\!04$$$$p^{72} T^{23} +$$$$27\!\cdots\!23$$$$p^{81} T^{24} -$$$$12\!\cdots\!30$$$$p^{90} T^{25} +$$$$72\!\cdots\!53$$$$p^{99} T^{26} -$$$$25\!\cdots\!76$$$$p^{108} T^{27} + 123939924002267915 p^{117} T^{28} - 247538102 p^{126} T^{29} + p^{135} T^{30}$$
67 $$1 - 467904656 T + 289992155760016797 T^{2} -$$$$96\!\cdots\!32$$$$T^{3} +$$$$36\!\cdots\!55$$$$T^{4} -$$$$99\!\cdots\!68$$$$T^{5} +$$$$29\!\cdots\!91$$$$T^{6} -$$$$69\!\cdots\!20$$$$T^{7} +$$$$17\!\cdots\!04$$$$T^{8} -$$$$36\!\cdots\!76$$$$T^{9} +$$$$81\!\cdots\!48$$$$T^{10} -$$$$15\!\cdots\!04$$$$T^{11} +$$$$31\!\cdots\!82$$$$T^{12} -$$$$54\!\cdots\!16$$$$T^{13} +$$$$10\!\cdots\!46$$$$T^{14} -$$$$16\!\cdots\!44$$$$T^{15} +$$$$10\!\cdots\!46$$$$p^{9} T^{16} -$$$$54\!\cdots\!16$$$$p^{18} T^{17} +$$$$31\!\cdots\!82$$$$p^{27} T^{18} -$$$$15\!\cdots\!04$$$$p^{36} T^{19} +$$$$81\!\cdots\!48$$$$p^{45} T^{20} -$$$$36\!\cdots\!76$$$$p^{54} T^{21} +$$$$17\!\cdots\!04$$$$p^{63} T^{22} -$$$$69\!\cdots\!20$$$$p^{72} T^{23} +$$$$29\!\cdots\!91$$$$p^{81} T^{24} -$$$$99\!\cdots\!68$$$$p^{90} T^{25} +$$$$36\!\cdots\!55$$$$p^{99} T^{26} -$$$$96\!\cdots\!32$$$$p^{108} T^{27} + 289992155760016797 p^{117} T^{28} - 467904656 p^{126} T^{29} + p^{135} T^{30}$$
71 $$1 + 8252944 T + 295175699032282861 T^{2} +$$$$11\!\cdots\!80$$$$T^{3} +$$$$45\!\cdots\!81$$$$T^{4} +$$$$31\!\cdots\!80$$$$T^{5} +$$$$48\!\cdots\!57$$$$T^{6} +$$$$47\!\cdots\!60$$$$T^{7} +$$$$40\!\cdots\!77$$$$T^{8} +$$$$47\!\cdots\!80$$$$T^{9} +$$$$28\!\cdots\!97$$$$T^{10} +$$$$35\!\cdots\!60$$$$T^{11} +$$$$17\!\cdots\!41$$$$T^{12} +$$$$20\!\cdots\!76$$$$T^{13} +$$$$94\!\cdots\!13$$$$T^{14} +$$$$10\!\cdots\!84$$$$T^{15} +$$$$94\!\cdots\!13$$$$p^{9} T^{16} +$$$$20\!\cdots\!76$$$$p^{18} T^{17} +$$$$17\!\cdots\!41$$$$p^{27} T^{18} +$$$$35\!\cdots\!60$$$$p^{36} T^{19} +$$$$28\!\cdots\!97$$$$p^{45} T^{20} +$$$$47\!\cdots\!80$$$$p^{54} T^{21} +$$$$40\!\cdots\!77$$$$p^{63} T^{22} +$$$$47\!\cdots\!60$$$$p^{72} T^{23} +$$$$48\!\cdots\!57$$$$p^{81} T^{24} +$$$$31\!\cdots\!80$$$$p^{90} T^{25} +$$$$45\!\cdots\!81$$$$p^{99} T^{26} +$$$$11\!\cdots\!80$$$$p^{108} T^{27} + 295175699032282861 p^{117} T^{28} + 8252944 p^{126} T^{29} + p^{135} T^{30}$$
73 $$1 + 715627902 T + 725896042339854143 T^{2} +$$$$39\!\cdots\!04$$$$T^{3} +$$$$23\!\cdots\!09$$$$T^{4} +$$$$10\!\cdots\!42$$$$T^{5} +$$$$48\!\cdots\!43$$$$T^{6} +$$$$18\!\cdots\!64$$$$T^{7} +$$$$69\!\cdots\!09$$$$T^{8} +$$$$23\!\cdots\!46$$$$T^{9} +$$$$77\!\cdots\!47$$$$T^{10} +$$$$23\!\cdots\!12$$$$T^{11} +$$$$68\!\cdots\!33$$$$T^{12} +$$$$18\!\cdots\!86$$$$T^{13} +$$$$49\!\cdots\!39$$$$T^{14} +$$$$11\!\cdots\!52$$$$T^{15} +$$$$49\!\cdots\!39$$$$p^{9} T^{16} +$$$$18\!\cdots\!86$$$$p^{18} T^{17} +$$$$68\!\cdots\!33$$$$p^{27} T^{18} +$$$$23\!\cdots\!12$$$$p^{36} T^{19} +$$$$77\!\cdots\!47$$$$p^{45} T^{20} +$$$$23\!\cdots\!46$$$$p^{54} T^{21} +$$$$69\!\cdots\!09$$$$p^{63} T^{22} +$$$$18\!\cdots\!64$$$$p^{72} T^{23} +$$$$48\!\cdots\!43$$$$p^{81} T^{24} +$$$$10\!\cdots\!42$$$$p^{90} T^{25} +$$$$23\!\cdots\!09$$$$p^{99} T^{26} +$$$$39\!\cdots\!04$$$$p^{108} T^{27} + 725896042339854143 p^{117} T^{28} + 715627902 p^{126} T^{29} + p^{135} T^{30}$$
79 $$1 - 560681783 T + 849133708768547829 T^{2} -$$$$32\!\cdots\!85$$$$T^{3} +$$$$33\!\cdots\!49$$$$T^{4} -$$$$10\!\cdots\!12$$$$T^{5} +$$$$90\!\cdots\!32$$$$T^{6} -$$$$23\!\cdots\!28$$$$T^{7} +$$$$19\!\cdots\!98$$$$T^{8} -$$$$43\!\cdots\!74$$$$T^{9} +$$$$34\!\cdots\!01$$$$T^{10} -$$$$67\!\cdots\!51$$$$T^{11} +$$$$53\!\cdots\!45$$$$T^{12} -$$$$93\!\cdots\!63$$$$T^{13} +$$$$72\!\cdots\!51$$$$T^{14} -$$$$11\!\cdots\!88$$$$T^{15} +$$$$72\!\cdots\!51$$$$p^{9} T^{16} -$$$$93\!\cdots\!63$$$$p^{18} T^{17} +$$$$53\!\cdots\!45$$$$p^{27} T^{18} -$$$$67\!\cdots\!51$$$$p^{36} T^{19} +$$$$34\!\cdots\!01$$$$p^{45} T^{20} -$$$$43\!\cdots\!74$$$$p^{54} T^{21} +$$$$19\!\cdots\!98$$$$p^{63} T^{22} -$$$$23\!\cdots\!28$$$$p^{72} T^{23} +$$$$90\!\cdots\!32$$$$p^{81} T^{24} -$$$$10\!\cdots\!12$$$$p^{90} T^{25} +$$$$33\!\cdots\!49$$$$p^{99} T^{26} -$$$$32\!\cdots\!85$$$$p^{108} T^{27} + 849133708768547829 p^{117} T^{28} - 560681783 p^{126} T^{29} + p^{135} T^{30}$$
83 $$1 + 1442854698 T + 2517006525957293097 T^{2} +$$$$25\!\cdots\!08$$$$T^{3} +$$$$27\!\cdots\!99$$$$T^{4} +$$$$22\!\cdots\!86$$$$T^{5} +$$$$19\!\cdots\!63$$$$T^{6} +$$$$13\!\cdots\!56$$$$T^{7} +$$$$92\!\cdots\!28$$$$T^{8} +$$$$56\!\cdots\!76$$$$T^{9} +$$$$33\!\cdots\!68$$$$T^{10} +$$$$18\!\cdots\!28$$$$T^{11} +$$$$97\!\cdots\!54$$$$T^{12} +$$$$47\!\cdots\!68$$$$T^{13} +$$$$22\!\cdots\!94$$$$T^{14} +$$$$98\!\cdots\!20$$$$T^{15} +$$$$22\!\cdots\!94$$$$p^{9} T^{16} +$$$$47\!\cdots\!68$$$$p^{18} T^{17} +$$$$97\!\cdots\!54$$$$p^{27} T^{18} +$$$$18\!\cdots\!28$$$$p^{36} T^{19} +$$$$33\!\cdots\!68$$$$p^{45} T^{20} +$$$$56\!\cdots\!76$$$$p^{54} T^{21} +$$$$92\!\cdots\!28$$$$p^{63} T^{22} +$$$$13\!\cdots\!56$$$$p^{72} T^{23} +$$$$19\!\cdots\!63$$$$p^{81} T^{24} +$$$$22\!\cdots\!86$$$$p^{90} T^{25} +$$$$27\!\cdots\!99$$$$p^{99} T^{26} +$$$$25\!\cdots\!08$$$$p^{108} T^{27} + 2517006525957293097 p^{117} T^{28} + 1442854698 p^{126} T^{29} + p^{135} T^{30}$$
89 $$1 + 396710008 T + 2452763342866776295 T^{2} +$$$$86\!\cdots\!68$$$$T^{3} +$$$$29\!\cdots\!85$$$$T^{4} +$$$$87\!\cdots\!92$$$$T^{5} +$$$$22\!\cdots\!31$$$$T^{6} +$$$$52\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!05$$$$T^{8} +$$$$17\!\cdots\!32$$$$T^{9} +$$$$51\!\cdots\!71$$$$T^{10} +$$$$14\!\cdots\!24$$$$T^{11} +$$$$17\!\cdots\!37$$$$T^{12} -$$$$18\!\cdots\!80$$$$T^{13} +$$$$54\!\cdots\!11$$$$T^{14} -$$$$10\!\cdots\!56$$$$T^{15} +$$$$54\!\cdots\!11$$$$p^{9} T^{16} -$$$$18\!\cdots\!80$$$$p^{18} T^{17} +$$$$17\!\cdots\!37$$$$p^{27} T^{18} +$$$$14\!\cdots\!24$$$$p^{36} T^{19} +$$$$51\!\cdots\!71$$$$p^{45} T^{20} +$$$$17\!\cdots\!32$$$$p^{54} T^{21} +$$$$12\!\cdots\!05$$$$p^{63} T^{22} +$$$$52\!\cdots\!68$$$$p^{72} T^{23} +$$$$22\!\cdots\!31$$$$p^{81} T^{24} +$$$$87\!\cdots\!92$$$$p^{90} T^{25} +$$$$29\!\cdots\!85$$$$p^{99} T^{26} +$$$$86\!\cdots\!68$$$$p^{108} T^{27} + 2452763342866776295 p^{117} T^{28} + 396710008 p^{126} T^{29} + p^{135} T^{30}$$
97 $$1 + 3063837815 T + 9536233785462653481 T^{2} +$$$$19\!\cdots\!71$$$$T^{3} +$$$$38\!\cdots\!99$$$$T^{4} +$$$$61\!\cdots\!48$$$$T^{5} +$$$$97\!\cdots\!08$$$$T^{6} +$$$$13\!\cdots\!04$$$$T^{7} +$$$$18\!\cdots\!19$$$$T^{8} +$$$$21\!\cdots\!95$$$$T^{9} +$$$$25\!\cdots\!96$$$$T^{10} +$$$$27\!\cdots\!24$$$$T^{11} +$$$$29\!\cdots\!54$$$$T^{12} +$$$$28\!\cdots\!39$$$$T^{13} +$$$$26\!\cdots\!31$$$$T^{14} +$$$$23\!\cdots\!02$$$$T^{15} +$$$$26\!\cdots\!31$$$$p^{9} T^{16} +$$$$28\!\cdots\!39$$$$p^{18} T^{17} +$$$$29\!\cdots\!54$$$$p^{27} T^{18} +$$$$27\!\cdots\!24$$$$p^{36} T^{19} +$$$$25\!\cdots\!96$$$$p^{45} T^{20} +$$$$21\!\cdots\!95$$$$p^{54} T^{21} +$$$$18\!\cdots\!19$$$$p^{63} T^{22} +$$$$13\!\cdots\!04$$$$p^{72} T^{23} +$$$$97\!\cdots\!08$$$$p^{81} T^{24} +$$$$61\!\cdots\!48$$$$p^{90} T^{25} +$$$$38\!\cdots\!99$$$$p^{99} T^{26} +$$$$19\!\cdots\!71$$$$p^{108} T^{27} + 9536233785462653481 p^{117} T^{28} + 3063837815 p^{126} T^{29} + p^{135} T^{30}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{30} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}