# Properties

 Degree $100$ Conductor $2.223\times 10^{163}$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $50$

# Learn more about

## Dirichlet series

 L(s)  = 1 − 10·2-s − 2·3-s − 57·4-s + 8·5-s + 20·6-s − 6·7-s + 884·8-s − 493·9-s − 80·10-s − 252·11-s + 114·12-s − 192·13-s + 60·14-s − 16·15-s + 748·16-s − 236·17-s + 4.93e3·18-s + 12·19-s − 456·20-s + 12·21-s + 2.52e3·22-s − 630·23-s − 1.76e3·24-s − 2.54e3·25-s + 1.92e3·26-s + 1.01e3·27-s + 342·28-s + ⋯
 L(s)  = 1 − 3.53·2-s − 0.384·3-s − 7.12·4-s + 0.715·5-s + 1.36·6-s − 0.323·7-s + 39.0·8-s − 18.2·9-s − 2.52·10-s − 6.90·11-s + 2.74·12-s − 4.09·13-s + 1.14·14-s − 0.275·15-s + 11.6·16-s − 3.36·17-s + 64.5·18-s + 0.144·19-s − 5.09·20-s + 0.124·21-s + 24.4·22-s − 5.71·23-s − 15.0·24-s − 20.3·25-s + 14.4·26-s + 7.19·27-s + 2.30·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{100}\right)^{s/2} \, \Gamma_{\C}(s)^{50} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{100}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{50} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$100$$ Conductor: $$43^{100}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{1849} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$50$$ Selberg data: $$(100,\ 43^{100} ,\ ( \ : [3/2]^{50} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1$$
good2 $$1 + 5 p T + 157 T^{2} + 157 p^{3} T^{3} + 11921 T^{4} + 20329 p^{2} T^{5} + 598259 T^{6} + 451937 p^{3} T^{7} + 2825747 p^{3} T^{8} + 31029871 p^{2} T^{9} + 690780623 T^{10} + 3506574575 T^{11} + 8925003671 p T^{12} + 42420347597 p T^{13} + 402020849581 T^{14} + 903079186685 p T^{15} + 8064201504339 T^{16} + 17248689251287 p T^{17} + 146412854050493 T^{18} + 599834624248145 T^{19} + 2436124011143725 T^{20} + 1200368290324333 p^{3} T^{21} + 37514071609118285 T^{22} + 35707058657518989 p^{2} T^{23} + 269456316845711465 p T^{24} + 124255534133014119 p^{4} T^{25} + 7269641615753078221 T^{26} + 26054958926883547857 T^{27} + 23146458155277925077 p^{2} T^{28} +$$$$16\!\cdots\!05$$$$p T^{29} +$$$$11\!\cdots\!31$$$$T^{30} +$$$$19\!\cdots\!01$$$$p T^{31} +$$$$64\!\cdots\!93$$$$p T^{32} +$$$$10\!\cdots\!59$$$$p^{2} T^{33} +$$$$70\!\cdots\!73$$$$p T^{34} +$$$$46\!\cdots\!67$$$$T^{35} +$$$$58\!\cdots\!13$$$$p^{8} T^{36} +$$$$29\!\cdots\!83$$$$p^{4} T^{37} +$$$$23\!\cdots\!89$$$$p^{6} T^{38} +$$$$14\!\cdots\!85$$$$p^{5} T^{39} +$$$$45\!\cdots\!69$$$$p^{5} T^{40} +$$$$35\!\cdots\!11$$$$p^{7} T^{41} +$$$$10\!\cdots\!43$$$$p^{7} T^{42} +$$$$16\!\cdots\!45$$$$p^{8} T^{43} +$$$$24\!\cdots\!01$$$$p^{9} T^{44} +$$$$35\!\cdots\!09$$$$p^{10} T^{45} +$$$$52\!\cdots\!21$$$$p^{11} T^{46} +$$$$76\!\cdots\!33$$$$p^{12} T^{47} +$$$$55\!\cdots\!11$$$$p^{14} T^{48} +$$$$39\!\cdots\!75$$$$p^{16} T^{49} +$$$$22\!\cdots\!31$$$$p^{15} T^{50} +$$$$39\!\cdots\!75$$$$p^{19} T^{51} +$$$$55\!\cdots\!11$$$$p^{20} T^{52} +$$$$76\!\cdots\!33$$$$p^{21} T^{53} +$$$$52\!\cdots\!21$$$$p^{23} T^{54} +$$$$35\!\cdots\!09$$$$p^{25} T^{55} +$$$$24\!\cdots\!01$$$$p^{27} T^{56} +$$$$16\!\cdots\!45$$$$p^{29} T^{57} +$$$$10\!\cdots\!43$$$$p^{31} T^{58} +$$$$35\!\cdots\!11$$$$p^{34} T^{59} +$$$$45\!\cdots\!69$$$$p^{35} T^{60} +$$$$14\!\cdots\!85$$$$p^{38} T^{61} +$$$$23\!\cdots\!89$$$$p^{42} T^{62} +$$$$29\!\cdots\!83$$$$p^{43} T^{63} +$$$$58\!\cdots\!13$$$$p^{50} T^{64} +$$$$46\!\cdots\!67$$$$p^{45} T^{65} +$$$$70\!\cdots\!73$$$$p^{49} T^{66} +$$$$10\!\cdots\!59$$$$p^{53} T^{67} +$$$$64\!\cdots\!93$$$$p^{55} T^{68} +$$$$19\!\cdots\!01$$$$p^{58} T^{69} +$$$$11\!\cdots\!31$$$$p^{60} T^{70} +$$$$16\!\cdots\!05$$$$p^{64} T^{71} + 23146458155277925077 p^{68} T^{72} + 26054958926883547857 p^{69} T^{73} + 7269641615753078221 p^{72} T^{74} + 124255534133014119 p^{79} T^{75} + 269456316845711465 p^{79} T^{76} + 35707058657518989 p^{83} T^{77} + 37514071609118285 p^{84} T^{78} + 1200368290324333 p^{90} T^{79} + 2436124011143725 p^{90} T^{80} + 599834624248145 p^{93} T^{81} + 146412854050493 p^{96} T^{82} + 17248689251287 p^{100} T^{83} + 8064201504339 p^{102} T^{84} + 903079186685 p^{106} T^{85} + 402020849581 p^{108} T^{86} + 42420347597 p^{112} T^{87} + 8925003671 p^{115} T^{88} + 3506574575 p^{117} T^{89} + 690780623 p^{120} T^{90} + 31029871 p^{125} T^{91} + 2825747 p^{129} T^{92} + 451937 p^{132} T^{93} + 598259 p^{132} T^{94} + 20329 p^{137} T^{95} + 11921 p^{138} T^{96} + 157 p^{144} T^{97} + 157 p^{144} T^{98} + 5 p^{148} T^{99} + p^{150} T^{100}$$
3 $$1 + 2 T + 497 T^{2} + 970 T^{3} + 125912 T^{4} + 248576 T^{5} + 21658841 T^{6} + 44433814 T^{7} + 2844186377 T^{8} + 6175504024 T^{9} + 101356113169 p T^{10} + 706183541222 T^{11} + 27569326270751 T^{12} + 68778648735704 T^{13} + 2180943544020203 T^{14} + 5841398931088804 T^{15} + 153702430892193628 T^{16} + 440243477738567258 T^{17} + 9804838095328353499 T^{18} + 29849870709928973030 T^{19} +$$$$57\!\cdots\!63$$$$T^{20} +$$$$18\!\cdots\!84$$$$T^{21} +$$$$10\!\cdots\!62$$$$p T^{22} +$$$$10\!\cdots\!58$$$$T^{23} +$$$$15\!\cdots\!35$$$$T^{24} +$$$$54\!\cdots\!20$$$$T^{25} +$$$$74\!\cdots\!69$$$$T^{26} +$$$$26\!\cdots\!46$$$$T^{27} +$$$$33\!\cdots\!45$$$$T^{28} +$$$$12\!\cdots\!90$$$$T^{29} +$$$$14\!\cdots\!87$$$$T^{30} +$$$$52\!\cdots\!08$$$$T^{31} +$$$$56\!\cdots\!91$$$$T^{32} +$$$$21\!\cdots\!94$$$$T^{33} +$$$$21\!\cdots\!82$$$$T^{34} +$$$$83\!\cdots\!40$$$$T^{35} +$$$$79\!\cdots\!65$$$$T^{36} +$$$$30\!\cdots\!94$$$$T^{37} +$$$$27\!\cdots\!29$$$$T^{38} +$$$$35\!\cdots\!00$$$$p T^{39} +$$$$93\!\cdots\!74$$$$T^{40} +$$$$11\!\cdots\!16$$$$p T^{41} +$$$$30\!\cdots\!80$$$$T^{42} +$$$$11\!\cdots\!28$$$$T^{43} +$$$$93\!\cdots\!68$$$$T^{44} +$$$$39\!\cdots\!82$$$$p^{2} T^{45} +$$$$27\!\cdots\!30$$$$T^{46} +$$$$34\!\cdots\!02$$$$p T^{47} +$$$$79\!\cdots\!06$$$$T^{48} +$$$$29\!\cdots\!10$$$$T^{49} +$$$$21\!\cdots\!64$$$$T^{50} +$$$$29\!\cdots\!10$$$$p^{3} T^{51} +$$$$79\!\cdots\!06$$$$p^{6} T^{52} +$$$$34\!\cdots\!02$$$$p^{10} T^{53} +$$$$27\!\cdots\!30$$$$p^{12} T^{54} +$$$$39\!\cdots\!82$$$$p^{17} T^{55} +$$$$93\!\cdots\!68$$$$p^{18} T^{56} +$$$$11\!\cdots\!28$$$$p^{21} T^{57} +$$$$30\!\cdots\!80$$$$p^{24} T^{58} +$$$$11\!\cdots\!16$$$$p^{28} T^{59} +$$$$93\!\cdots\!74$$$$p^{30} T^{60} +$$$$35\!\cdots\!00$$$$p^{34} T^{61} +$$$$27\!\cdots\!29$$$$p^{36} T^{62} +$$$$30\!\cdots\!94$$$$p^{39} T^{63} +$$$$79\!\cdots\!65$$$$p^{42} T^{64} +$$$$83\!\cdots\!40$$$$p^{45} T^{65} +$$$$21\!\cdots\!82$$$$p^{48} T^{66} +$$$$21\!\cdots\!94$$$$p^{51} T^{67} +$$$$56\!\cdots\!91$$$$p^{54} T^{68} +$$$$52\!\cdots\!08$$$$p^{57} T^{69} +$$$$14\!\cdots\!87$$$$p^{60} T^{70} +$$$$12\!\cdots\!90$$$$p^{63} T^{71} +$$$$33\!\cdots\!45$$$$p^{66} T^{72} +$$$$26\!\cdots\!46$$$$p^{69} T^{73} +$$$$74\!\cdots\!69$$$$p^{72} T^{74} +$$$$54\!\cdots\!20$$$$p^{75} T^{75} +$$$$15\!\cdots\!35$$$$p^{78} T^{76} +$$$$10\!\cdots\!58$$$$p^{81} T^{77} +$$$$10\!\cdots\!62$$$$p^{85} T^{78} +$$$$18\!\cdots\!84$$$$p^{87} T^{79} +$$$$57\!\cdots\!63$$$$p^{90} T^{80} + 29849870709928973030 p^{93} T^{81} + 9804838095328353499 p^{96} T^{82} + 440243477738567258 p^{99} T^{83} + 153702430892193628 p^{102} T^{84} + 5841398931088804 p^{105} T^{85} + 2180943544020203 p^{108} T^{86} + 68778648735704 p^{111} T^{87} + 27569326270751 p^{114} T^{88} + 706183541222 p^{117} T^{89} + 101356113169 p^{121} T^{90} + 6175504024 p^{123} T^{91} + 2844186377 p^{126} T^{92} + 44433814 p^{129} T^{93} + 21658841 p^{132} T^{94} + 248576 p^{135} T^{95} + 125912 p^{138} T^{96} + 970 p^{141} T^{97} + 497 p^{144} T^{98} + 2 p^{147} T^{99} + p^{150} T^{100}$$
5 $$1 - 8 T + 2608 T^{2} - 21134 T^{3} + 3416438 T^{4} - 27631954 T^{5} + 2998814584 T^{6} - 23907916402 T^{7} + 1984951570099 T^{8} - 15435963396718 T^{9} + 42289305581151 p^{2} T^{10} - 1589650997636372 p T^{11} + 472223820950978032 T^{12} - 3405957509564661396 T^{13} +$$$$18\!\cdots\!74$$$$T^{14} -$$$$12\!\cdots\!84$$$$T^{15} +$$$$61\!\cdots\!98$$$$T^{16} -$$$$40\!\cdots\!66$$$$T^{17} +$$$$75\!\cdots\!46$$$$p^{2} T^{18} -$$$$11\!\cdots\!18$$$$T^{19} +$$$$52\!\cdots\!71$$$$T^{20} -$$$$60\!\cdots\!48$$$$p T^{21} +$$$$13\!\cdots\!09$$$$T^{22} -$$$$71\!\cdots\!72$$$$T^{23} +$$$$31\!\cdots\!01$$$$T^{24} -$$$$15\!\cdots\!36$$$$T^{25} +$$$$68\!\cdots\!28$$$$T^{26} -$$$$31\!\cdots\!96$$$$T^{27} +$$$$14\!\cdots\!72$$$$T^{28} -$$$$60\!\cdots\!34$$$$T^{29} +$$$$55\!\cdots\!31$$$$p T^{30} -$$$$21\!\cdots\!82$$$$p T^{31} +$$$$50\!\cdots\!51$$$$T^{32} -$$$$18\!\cdots\!22$$$$T^{33} +$$$$17\!\cdots\!56$$$$p T^{34} -$$$$30\!\cdots\!86$$$$T^{35} +$$$$14\!\cdots\!97$$$$T^{36} -$$$$47\!\cdots\!26$$$$T^{37} +$$$$23\!\cdots\!13$$$$T^{38} -$$$$14\!\cdots\!74$$$$p T^{39} +$$$$36\!\cdots\!46$$$$T^{40} -$$$$10\!\cdots\!72$$$$T^{41} +$$$$10\!\cdots\!17$$$$p T^{42} -$$$$14\!\cdots\!62$$$$T^{43} +$$$$15\!\cdots\!89$$$$p T^{44} -$$$$18\!\cdots\!82$$$$T^{45} +$$$$10\!\cdots\!08$$$$T^{46} -$$$$99\!\cdots\!06$$$$p^{2} T^{47} +$$$$13\!\cdots\!87$$$$T^{48} -$$$$31\!\cdots\!62$$$$T^{49} +$$$$17\!\cdots\!99$$$$T^{50} -$$$$31\!\cdots\!62$$$$p^{3} T^{51} +$$$$13\!\cdots\!87$$$$p^{6} T^{52} -$$$$99\!\cdots\!06$$$$p^{11} T^{53} +$$$$10\!\cdots\!08$$$$p^{12} T^{54} -$$$$18\!\cdots\!82$$$$p^{15} T^{55} +$$$$15\!\cdots\!89$$$$p^{19} T^{56} -$$$$14\!\cdots\!62$$$$p^{21} T^{57} +$$$$10\!\cdots\!17$$$$p^{25} T^{58} -$$$$10\!\cdots\!72$$$$p^{27} T^{59} +$$$$36\!\cdots\!46$$$$p^{30} T^{60} -$$$$14\!\cdots\!74$$$$p^{34} T^{61} +$$$$23\!\cdots\!13$$$$p^{36} T^{62} -$$$$47\!\cdots\!26$$$$p^{39} T^{63} +$$$$14\!\cdots\!97$$$$p^{42} T^{64} -$$$$30\!\cdots\!86$$$$p^{45} T^{65} +$$$$17\!\cdots\!56$$$$p^{49} T^{66} -$$$$18\!\cdots\!22$$$$p^{51} T^{67} +$$$$50\!\cdots\!51$$$$p^{54} T^{68} -$$$$21\!\cdots\!82$$$$p^{58} T^{69} +$$$$55\!\cdots\!31$$$$p^{61} T^{70} -$$$$60\!\cdots\!34$$$$p^{63} T^{71} +$$$$14\!\cdots\!72$$$$p^{66} T^{72} -$$$$31\!\cdots\!96$$$$p^{69} T^{73} +$$$$68\!\cdots\!28$$$$p^{72} T^{74} -$$$$15\!\cdots\!36$$$$p^{75} T^{75} +$$$$31\!\cdots\!01$$$$p^{78} T^{76} -$$$$71\!\cdots\!72$$$$p^{81} T^{77} +$$$$13\!\cdots\!09$$$$p^{84} T^{78} -$$$$60\!\cdots\!48$$$$p^{88} T^{79} +$$$$52\!\cdots\!71$$$$p^{90} T^{80} -$$$$11\!\cdots\!18$$$$p^{93} T^{81} +$$$$75\!\cdots\!46$$$$p^{98} T^{82} -$$$$40\!\cdots\!66$$$$p^{99} T^{83} +$$$$61\!\cdots\!98$$$$p^{102} T^{84} -$$$$12\!\cdots\!84$$$$p^{105} T^{85} +$$$$18\!\cdots\!74$$$$p^{108} T^{86} - 3405957509564661396 p^{111} T^{87} + 472223820950978032 p^{114} T^{88} - 1589650997636372 p^{118} T^{89} + 42289305581151 p^{122} T^{90} - 15435963396718 p^{123} T^{91} + 1984951570099 p^{126} T^{92} - 23907916402 p^{129} T^{93} + 2998814584 p^{132} T^{94} - 27631954 p^{135} T^{95} + 3416438 p^{138} T^{96} - 21134 p^{141} T^{97} + 2608 p^{144} T^{98} - 8 p^{147} T^{99} + p^{150} T^{100}$$
7 $$1 + 6 T + 1133 p T^{2} + 43816 T^{3} + 31283732 T^{4} + 163985858 T^{5} + 81970791283 T^{6} + 421162311562 T^{7} + 160754991367878 T^{8} + 834768402128408 T^{9} + 252030845068943587 T^{10} + 193731135781886118 p T^{11} +$$$$32\!\cdots\!94$$$$T^{12} +$$$$18\!\cdots\!38$$$$T^{13} +$$$$36\!\cdots\!60$$$$T^{14} +$$$$22\!\cdots\!94$$$$T^{15} +$$$$36\!\cdots\!70$$$$T^{16} +$$$$23\!\cdots\!22$$$$T^{17} +$$$$31\!\cdots\!47$$$$T^{18} +$$$$21\!\cdots\!30$$$$T^{19} +$$$$25\!\cdots\!02$$$$T^{20} +$$$$18\!\cdots\!32$$$$T^{21} +$$$$18\!\cdots\!09$$$$T^{22} +$$$$13\!\cdots\!58$$$$T^{23} +$$$$35\!\cdots\!29$$$$p^{3} T^{24} +$$$$95\!\cdots\!18$$$$T^{25} +$$$$10\!\cdots\!20$$$$p T^{26} +$$$$60\!\cdots\!92$$$$T^{27} +$$$$44\!\cdots\!06$$$$T^{28} +$$$$35\!\cdots\!24$$$$T^{29} +$$$$24\!\cdots\!51$$$$T^{30} +$$$$27\!\cdots\!56$$$$p T^{31} +$$$$25\!\cdots\!37$$$$p^{2} T^{32} +$$$$98\!\cdots\!42$$$$T^{33} +$$$$60\!\cdots\!35$$$$T^{34} +$$$$47\!\cdots\!26$$$$T^{35} +$$$$27\!\cdots\!01$$$$T^{36} +$$$$21\!\cdots\!78$$$$T^{37} +$$$$12\!\cdots\!17$$$$T^{38} +$$$$90\!\cdots\!38$$$$T^{39} +$$$$51\!\cdots\!56$$$$T^{40} +$$$$36\!\cdots\!14$$$$T^{41} +$$$$20\!\cdots\!77$$$$T^{42} +$$$$14\!\cdots\!78$$$$T^{43} +$$$$80\!\cdots\!77$$$$T^{44} +$$$$54\!\cdots\!28$$$$T^{45} +$$$$30\!\cdots\!96$$$$T^{46} +$$$$19\!\cdots\!42$$$$T^{47} +$$$$10\!\cdots\!91$$$$T^{48} +$$$$69\!\cdots\!34$$$$T^{49} +$$$$38\!\cdots\!70$$$$T^{50} +$$$$69\!\cdots\!34$$$$p^{3} T^{51} +$$$$10\!\cdots\!91$$$$p^{6} T^{52} +$$$$19\!\cdots\!42$$$$p^{9} T^{53} +$$$$30\!\cdots\!96$$$$p^{12} T^{54} +$$$$54\!\cdots\!28$$$$p^{15} T^{55} +$$$$80\!\cdots\!77$$$$p^{18} T^{56} +$$$$14\!\cdots\!78$$$$p^{21} T^{57} +$$$$20\!\cdots\!77$$$$p^{24} T^{58} +$$$$36\!\cdots\!14$$$$p^{27} T^{59} +$$$$51\!\cdots\!56$$$$p^{30} T^{60} +$$$$90\!\cdots\!38$$$$p^{33} T^{61} +$$$$12\!\cdots\!17$$$$p^{36} T^{62} +$$$$21\!\cdots\!78$$$$p^{39} T^{63} +$$$$27\!\cdots\!01$$$$p^{42} T^{64} +$$$$47\!\cdots\!26$$$$p^{45} T^{65} +$$$$60\!\cdots\!35$$$$p^{48} T^{66} +$$$$98\!\cdots\!42$$$$p^{51} T^{67} +$$$$25\!\cdots\!37$$$$p^{56} T^{68} +$$$$27\!\cdots\!56$$$$p^{58} T^{69} +$$$$24\!\cdots\!51$$$$p^{60} T^{70} +$$$$35\!\cdots\!24$$$$p^{63} T^{71} +$$$$44\!\cdots\!06$$$$p^{66} T^{72} +$$$$60\!\cdots\!92$$$$p^{69} T^{73} +$$$$10\!\cdots\!20$$$$p^{73} T^{74} +$$$$95\!\cdots\!18$$$$p^{75} T^{75} +$$$$35\!\cdots\!29$$$$p^{81} T^{76} +$$$$13\!\cdots\!58$$$$p^{81} T^{77} +$$$$18\!\cdots\!09$$$$p^{84} T^{78} +$$$$18\!\cdots\!32$$$$p^{87} T^{79} +$$$$25\!\cdots\!02$$$$p^{90} T^{80} +$$$$21\!\cdots\!30$$$$p^{93} T^{81} +$$$$31\!\cdots\!47$$$$p^{96} T^{82} +$$$$23\!\cdots\!22$$$$p^{99} T^{83} +$$$$36\!\cdots\!70$$$$p^{102} T^{84} +$$$$22\!\cdots\!94$$$$p^{105} T^{85} +$$$$36\!\cdots\!60$$$$p^{108} T^{86} +$$$$18\!\cdots\!38$$$$p^{111} T^{87} +$$$$32\!\cdots\!94$$$$p^{114} T^{88} + 193731135781886118 p^{118} T^{89} + 252030845068943587 p^{120} T^{90} + 834768402128408 p^{123} T^{91} + 160754991367878 p^{126} T^{92} + 421162311562 p^{129} T^{93} + 81970791283 p^{132} T^{94} + 163985858 p^{135} T^{95} + 31283732 p^{138} T^{96} + 43816 p^{141} T^{97} + 1133 p^{145} T^{98} + 6 p^{147} T^{99} + p^{150} T^{100}$$
11 $$1 + 252T + 6.41e4T^{2} + 1.08e7T^{3} + 1.71e9T^{4} + 2.25e11T^{5} + 2.78e13T^{6} + 3.06e15T^{7} + 3.19e17T^{8} + 3.06e19T^{9} + 2.80e21T^{10} + 2.41e23T^{11} + 1.98e25T^{12} + 1.55e27T^{13} + 1.17e29T^{14} + 8.49e30T^{15} + 5.94e32T^{16} + 4.01e34T^{17} + 2.62e36T^{18} + 1.66e38T^{19} + 1.02e40T^{20} + 6.16e41T^{21} + 3.60e43T^{22} + 2.05e45T^{23} + 1.14e47T^{24} + 6.23e48T^{25} + 3.32e50T^{26} + 1.73e52T^{27} + 8.86e53T^{28} + 4.44e55T^{29} + 2.18e57T^{30} + 1.05e59T^{31} + 5.01e60T^{32} + 2.33e62T^{33} + 1.07e64T^{34} + 4.84e65T^{35} + 2.15e67T^{36} + 9.43e68T^{37} + 4.07e70T^{38} + 1.73e72T^{39} + 7.25e73T^{40} + 2.99e75T^{41} + 1.22e77T^{42} + 4.91e78T^{43} + 1.95e80T^{44} + 7.64e81T^{45} + 2.95e83T^{46} + 1.12e85T^{47}+O(T^{48})$$
13 $$1 + 192T + 6.76e4T^{2} + 9.93e6T^{3} + 2.05e9T^{4} + 2.49e11T^{5} + 3.90e13T^{6} + 4.07e15T^{7} + 5.30e17T^{8} + 4.88e19T^{9} + 5.57e21T^{10} + 4.62e23T^{11} + 4.78e25T^{12} + 3.60e27T^{13} + 3.45e29T^{14} + 2.39e31T^{15} + 2.16e33T^{16} + 1.38e35T^{17} + 1.20e37T^{18} + 7.14e38T^{19} + 6.01e40T^{20} + 3.30e42T^{21} + 2.73e44T^{22} + 1.39e46T^{23} + 1.15e48T^{24} + 5.43e49T^{25} + 4.49e51T^{26} + 1.95e53T^{27} + 1.64e55T^{28} + 6.58e56T^{29} + 5.66e58T^{30} + 2.07e60T^{31} + 1.85e62T^{32} + 6.18e63T^{33} + 5.74e65T^{34} + 1.74e67T^{35} + 1.70e69T^{36} + 4.67e70T^{37} + 4.82e72T^{38} + 1.19e74T^{39} + 1.31e76T^{40} + 2.95e77T^{41} + 3.42e79T^{42} + 7.03e80T^{43} + 8.57e82T^{44}+O(T^{45})$$
17 $$1 + 236T + 1.57e5T^{2} + 3.18e7T^{3} + 1.19e10T^{4} + 2.12e12T^{5} + 5.81e14T^{6} + 9.33e16T^{7} + 2.08e19T^{8} + 3.05e21T^{9} + 5.85e23T^{10} + 7.94e25T^{11} + 1.34e28T^{12} + 1.70e30T^{13} + 2.62e32T^{14} + 3.11e34T^{15} + 4.40e36T^{16} + 4.94e38T^{17} + 6.49e40T^{18} + 6.92e42T^{19} + 8.52e44T^{20} + 8.64e46T^{21} + 1.00e49T^{22} + 9.73e50T^{23} + 1.07e53T^{24} + 9.94e54T^{25} + 1.04e57T^{26} + 9.29e58T^{27} + 9.33e60T^{28} + 7.98e62T^{29} + 7.69e64T^{30} + 6.33e66T^{31} + 5.87e68T^{32} + 4.65e70T^{33} + 4.17e72T^{34} + 3.19e74T^{35} + 2.76e76T^{36} + 2.04e78T^{37} + 1.72e80T^{38} + 1.23e82T^{39} + 1.01e84T^{40}+O(T^{41})$$
19 $$1 - 12T + 1.73e5T^{2} - 2.51e6T^{3} + 1.51e10T^{4} - 2.53e11T^{5} + 8.81e14T^{6} - 1.66e16T^{7} + 3.87e19T^{8} - 8.03e20T^{9} + 1.36e24T^{10} - 3.06e25T^{11} + 4.05e28T^{12} - 9.69e29T^{13} + 1.03e33T^{14} - 2.60e34T^{15} + 2.31e37T^{16} - 6.11e38T^{17} + 4.63e41T^{18} - 1.27e43T^{19} + 8.37e45T^{20} - 2.36e47T^{21} + 1.38e50T^{22} - 4.00e51T^{23} + 2.09e54T^{24} - 6.20e55T^{25} + 2.93e58T^{26} - 8.84e59T^{27} + 3.83e62T^{28} - 1.16e64T^{29} + 4.67e66T^{30} - 1.43e68T^{31} + 5.34e70T^{32} - 1.65e72T^{33} + 5.75e74T^{34} - 1.77e76T^{35} + 5.84e78T^{36} - 1.80e80T^{37} + 5.60e82T^{38}+O(T^{39})$$
23 $$1 + 630T + 4.62e5T^{2} + 2.04e8T^{3} + 9.22e10T^{4} + 3.25e13T^{5} + 1.13e16T^{6} + 3.40e18T^{7} + 1.00e21T^{8} + 2.64e23T^{9} + 6.83e25T^{10} + 1.63e28T^{11} + 3.81e30T^{12} + 8.35e32T^{13} + 1.79e35T^{14} + 3.64e37T^{15} + 7.28e39T^{16} + 1.39e42T^{17} + 2.60e44T^{18} + 4.69e46T^{19} + 8.32e48T^{20} + 1.42e51T^{21} + 2.40e53T^{22} + 3.93e55T^{23} + 6.33e57T^{24} + 9.93e59T^{25} + 1.53e62T^{26} + 2.31e64T^{27} + 3.44e66T^{28} + 5.00e68T^{29} + 7.19e70T^{30} + 1.01e73T^{31} + 1.40e75T^{32} + 1.91e77T^{33} + 2.58e79T^{34} + 3.43e81T^{35} + 4.49e83T^{36}+O(T^{37})$$
29 $$1 - 208T + 6.27e5T^{2} - 1.37e8T^{3} + 1.98e11T^{4} - 4.51e13T^{5} + 4.19e16T^{6} - 9.84e18T^{7} + 6.70e21T^{8} - 1.60e24T^{9} + 8.62e26T^{10} - 2.08e29T^{11} + 9.29e31T^{12} - 2.25e34T^{13} + 8.63e36T^{14} - 2.08e39T^{15} + 7.05e41T^{16} - 1.68e44T^{17} + 5.14e46T^{18} - 1.21e49T^{19} + 3.38e51T^{20} - 7.80e53T^{21} + 2.03e56T^{22} - 4.57e58T^{23} + 1.11e61T^{24} - 2.45e63T^{25} + 5.68e65T^{26} - 1.21e68T^{27} + 2.68e70T^{28} - 5.56e72T^{29} + 1.17e75T^{30} - 2.37e77T^{31} + 4.85e79T^{32} - 9.52e81T^{33} + 1.87e84T^{34}+O(T^{35})$$
31 $$1 + 932T + 1.19e6T^{2} + 8.27e8T^{3} + 6.35e11T^{4} + 3.58e14T^{5} + 2.10e17T^{6} + 1.01e20T^{7} + 4.97e22T^{8} + 2.12e25T^{9} + 9.10e27T^{10} + 3.51e30T^{11} + 1.35e33T^{12} + 4.80e35T^{13} + 1.69e38T^{14} + 5.57e40T^{15} + 1.82e43T^{16} + 5.63e45T^{17} + 1.72e48T^{18} + 5.03e50T^{19} + 1.45e53T^{20} + 4.03e55T^{21} + 1.10e58T^{22} + 2.93e60T^{23} + 7.68e62T^{24} + 1.94e65T^{25} + 4.89e67T^{26} + 1.19e70T^{27} + 2.88e72T^{28} + 6.77e74T^{29} + 1.57e77T^{30} + 3.57e79T^{31} + 8.05e81T^{32}+O(T^{33})$$
37 $$1 + 90T + 1.32e6T^{2} + 1.60e8T^{3} + 8.78e11T^{4} + 1.32e14T^{5} + 3.92e17T^{6} + 7.00e19T^{7} + 1.32e23T^{8} + 2.69e25T^{9} + 3.63e28T^{10} + 8.13e30T^{11} + 8.36e33T^{12} + 2.01e36T^{13} + 1.66e39T^{14} + 4.24e41T^{15} + 2.92e44T^{16} + 7.76e46T^{17} + 4.60e49T^{18} + 1.25e52T^{19} + 6.56e54T^{20} + 1.81e57T^{21} + 8.53e59T^{22} + 2.36e62T^{23} + 1.01e65T^{24} + 2.82e67T^{25} + 1.12e70T^{26} + 3.10e72T^{27} + 1.15e75T^{28} + 3.14e77T^{29} + 1.09e80T^{30} + 2.95e82T^{31}+O(T^{32})$$
41 $$1 + 1.35e3T + 2.73e6T^{2} + 2.87e9T^{3} + 3.45e12T^{4} + 3.00e15T^{5} + 2.74e18T^{6} + 2.06e21T^{7} + 1.57e24T^{8} + 1.05e27T^{9} + 6.98e29T^{10} + 4.22e32T^{11} + 2.51e35T^{12} + 1.39e38T^{13} + 7.58e40T^{14} + 3.88e43T^{15} + 1.95e46T^{16} + 9.38e48T^{17} + 4.41e51T^{18} + 1.98e54T^{19} + 8.80e56T^{20} + 3.75e59T^{21} + 1.57e62T^{22} + 6.36e64T^{23} + 2.53e67T^{24} + 9.79e69T^{25} + 3.72e72T^{26} + 1.37e75T^{27} + 5.02e77T^{28} + 1.78e80T^{29} + 6.25e82T^{30}+O(T^{31})$$
47 $$1 + 3.48e3T + 7.90e6T^{2} + 1.32e10T^{3} + 1.83e13T^{4} + 2.17e16T^{5} + 2.29e19T^{6} + 2.17e22T^{7} + 1.89e25T^{8} + 1.53e28T^{9} + 1.16e31T^{10} + 8.31e33T^{11} + 5.63e36T^{12} + 3.64e39T^{13} + 2.26e42T^{14} + 1.34e45T^{15} + 7.74e47T^{16} + 4.30e50T^{17} + 2.32e53T^{18} + 1.21e56T^{19} + 6.20e58T^{20} + 3.08e61T^{21} + 1.49e64T^{22} + 7.11e66T^{23} + 3.30e69T^{24} + 1.50e72T^{25} + 6.73e74T^{26} + 2.95e77T^{27} + 1.27e80T^{28} + 5.39e82T^{29}+O(T^{30})$$
53 $$1 + 726T + 3.70e6T^{2} + 2.65e9T^{3} + 6.94e12T^{4} + 4.87e15T^{5} + 8.73e18T^{6} + 5.97e21T^{7} + 8.28e24T^{8} + 5.49e27T^{9} + 6.30e30T^{10} + 4.04e33T^{11} + 3.99e36T^{12} + 2.47e39T^{13} + 2.16e42T^{14} + 1.29e45T^{15} + 1.02e48T^{16} + 5.91e50T^{17} + 4.30e53T^{18} + 2.39e56T^{19} + 1.61e59T^{20} + 8.67e61T^{21} + 5.48e64T^{22} + 2.84e67T^{23} + 1.69e70T^{24} + 8.51e72T^{25} + 4.82e75T^{26} + 2.34e78T^{27} + 1.26e81T^{28}+O(T^{29})$$
59 $$1 + 4.37e3T + 1.44e7T^{2} + 3.50e10T^{3} + 7.29e13T^{4} + 1.30e17T^{5} + 2.09e20T^{6} + 3.05e23T^{7} + 4.10e26T^{8} + 5.13e29T^{9} + 6.03e32T^{10} + 6.68e35T^{11} + 7.03e38T^{12} + 7.05e41T^{13} + 6.76e44T^{14} + 6.21e47T^{15} + 5.49e50T^{16} + 4.68e53T^{17} + 3.85e56T^{18} + 3.07e59T^{19} + 2.37e62T^{20} + 1.78e65T^{21} + 1.29e68T^{22} + 9.22e70T^{23} + 6.37e73T^{24} + 4.29e76T^{25} + 2.82e79T^{26} + 1.81e82T^{27}+O(T^{28})$$
61 $$1 - 1.17e3T + 5.52e6T^{2} - 5.87e9T^{3} + 1.50e13T^{4} - 1.46e16T^{5} + 2.69e19T^{6} - 2.41e22T^{7} + 3.57e25T^{8} - 2.97e28T^{9} + 3.75e31T^{10} - 2.91e34T^{11} + 3.24e37T^{12} - 2.35e40T^{13} + 2.38e43T^{14} - 1.62e46T^{15} + 1.51e49T^{16} - 9.79e51T^{17} + 8.52e54T^{18} - 5.21e57T^{19} + 4.28e60T^{20} - 2.48e63T^{21} + 1.95e66T^{22} - 1.07e69T^{23} + 8.11e71T^{24} - 4.25e74T^{25} + 3.11e77T^{26} - 1.55e80T^{27}+O(T^{28})$$
67 $$1 + 344T + 7.03e6T^{2} + 2.04e9T^{3} + 2.45e13T^{4} + 5.82e15T^{5} + 5.66e19T^{6} + 1.05e22T^{7} + 9.75e25T^{8} + 1.33e28T^{9} + 1.34e32T^{10} + 1.18e34T^{11} + 1.53e38T^{12} + 6.41e39T^{13} + 1.51e44T^{14} - 3.88e44T^{15} + 1.30e50T^{16} - 5.80e51T^{17} + 1.00e56T^{18} - 8.43e57T^{19} + 7.04e61T^{20} - 8.45e63T^{21} + 4.50e67T^{22} - 6.91e69T^{23} + 2.66e73T^{24} - 4.89e75T^{25} + 1.46e79T^{26} - 3.08e81T^{27}+O(T^{28})$$
71 $$1 - 162T + 9.76e6T^{2} - 2.19e9T^{3} + 4.75e13T^{4} - 1.35e16T^{5} + 1.54e20T^{6} - 5.27e22T^{7} + 3.75e26T^{8} - 1.48e29T^{9} + 7.33e32T^{10} - 3.25e35T^{11} + 1.19e39T^{12} - 5.83e41T^{13} + 1.67e45T^{14} - 8.81e47T^{15} + 2.05e51T^{16} - 1.15e54T^{17} + 2.25e57T^{18} - 1.32e60T^{19} + 2.22e63T^{20} - 1.35e66T^{21} + 2.00e69T^{22} - 1.25e72T^{23} + 1.65e75T^{24} - 1.05e78T^{25} + 1.26e81T^{26}+O(T^{27})$$
73 $$1 - 746T + 1.04e7T^{2} - 7.52e9T^{3} + 5.39e13T^{4} - 3.75e16T^{5} + 1.83e20T^{6} - 1.23e23T^{7} + 4.64e26T^{8} - 3.03e29T^{9} + 9.29e32T^{10} - 5.87e35T^{11} + 1.53e39T^{12} - 9.36e41T^{13} + 2.14e45T^{14} - 1.26e48T^{15} + 2.59e51T^{16} - 1.48e54T^{17} + 2.76e57T^{18} - 1.52e60T^{19} + 2.62e63T^{20} - 1.38e66T^{21} + 2.24e69T^{22} - 1.13e72T^{23} + 1.73e75T^{24} - 8.35e77T^{25} + 1.23e81T^{26}+O(T^{27})$$
79 $$1 + 2.65e3T + 1.62e7T^{2} + 3.70e10T^{3} + 1.28e14T^{4} + 2.58e17T^{5} + 6.62e20T^{6} + 1.20e24T^{7} + 2.54e27T^{8} + 4.24e30T^{9} + 7.72e33T^{10} + 1.19e37T^{11} + 1.94e40T^{12} + 2.80e43T^{13} + 4.16e46T^{14} + 5.64e49T^{15} + 7.76e52T^{16} + 9.93e55T^{17} + 1.27e59T^{18} + 1.55e62T^{19} + 1.88e65T^{20} + 2.18e68T^{21} + 2.51e71T^{22} + 2.78e74T^{23} + 3.05e77T^{24} + 3.24e80T^{25} + 3.41e83T^{26}+O(T^{27})$$
83 $$1 + 3.51e3T + 2.24e7T^{2} + 6.43e10T^{3} + 2.38e14T^{4} + 5.83e17T^{5} + 1.62e21T^{6} + 3.49e24T^{7} + 8.04e27T^{8} + 1.55e31T^{9} + 3.11e34T^{10} + 5.49e37T^{11} + 9.83e40T^{12} + 1.60e44T^{13} + 2.61e47T^{14} + 3.96e50T^{15} + 5.99e53T^{16} + 8.51e56T^{17} + 1.20e60T^{18} + 1.61e63T^{19} + 2.14e66T^{20} + 2.72e69T^{21} + 3.42e72T^{22} + 4.14e75T^{23} + 4.95e78T^{24} + 5.73e81T^{25}+O(T^{26})$$
89 $$1 - 2.64e3T + 1.97e7T^{2} - 4.32e10T^{3} + 1.84e14T^{4} - 3.49e17T^{5} + 1.10e21T^{6} - 1.86e24T^{7} + 4.86e27T^{8} - 7.49e30T^{9} + 1.69e34T^{10} - 2.40e37T^{11} + 4.87e40T^{12} - 6.46e43T^{13} + 1.20e47T^{14} - 1.49e50T^{15} + 2.58e53T^{16} - 3.06e56T^{17} + 4.96e59T^{18} - 5.60e62T^{19} + 8.60e65T^{20} - 9.31e68T^{21} + 1.36e72T^{22} - 1.41e75T^{23} + 1.98e78T^{24} - 1.99e81T^{25}+O(T^{26})$$
97 $$1 + 3.86e3T + 3.25e7T^{2} + 1.04e11T^{3} + 5.01e14T^{4} + 1.38e18T^{5} + 4.93e21T^{6} + 1.20e25T^{7} + 3.52e28T^{8} + 7.77e31T^{9} + 1.96e35T^{10} + 3.95e38T^{11} + 8.90e41T^{12} + 1.65e45T^{13} + 3.38e48T^{14} + 5.88e51T^{15} + 1.10e55T^{16} + 1.80e58T^{17} + 3.15e61T^{18} + 4.83e64T^{19} + 7.93e67T^{20} + 1.15e71T^{21} + 1.78e74T^{22} + 2.45e77T^{23} + 3.60e80T^{24}+O(T^{25})$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{100} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−1.25288227611160936432662562871, −1.23812506015909785015469151298, −1.23657124882801991139565804877, −1.21227170392721501581045563924, −1.17665603108512355394168117179, −1.13547548158704060747843006525, −1.03394619289072975829761352447, −1.01245461134464940880511470420, −0.998814198239016407257136529250, −0.976167151364885328108151243550, −0.970975010203764106289884219143, −0.959372897747489400776356758491, −0.954804614203070909673691101277, −0.954184414233699478199482101069, −0.925377485845133948859926572905, −0.864469026376641852906529146228, −0.792991629104931123358059368856, −0.77813855566844007094210072974, −0.75193275056024517399436821596, −0.74271620810225188644116146328, −0.70383259626875231992542951772, −0.67266851786011116871264386713, −0.67073495555897230798352543142, −0.61406931325781491862928900344, −0.41663249541034762289167925040, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.41663249541034762289167925040, 0.61406931325781491862928900344, 0.67073495555897230798352543142, 0.67266851786011116871264386713, 0.70383259626875231992542951772, 0.74271620810225188644116146328, 0.75193275056024517399436821596, 0.77813855566844007094210072974, 0.792991629104931123358059368856, 0.864469026376641852906529146228, 0.925377485845133948859926572905, 0.954184414233699478199482101069, 0.954804614203070909673691101277, 0.959372897747489400776356758491, 0.970975010203764106289884219143, 0.976167151364885328108151243550, 0.998814198239016407257136529250, 1.01245461134464940880511470420, 1.03394619289072975829761352447, 1.13547548158704060747843006525, 1.17665603108512355394168117179, 1.21227170392721501581045563924, 1.23657124882801991139565804877, 1.23812506015909785015469151298, 1.25288227611160936432662562871

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

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