# Properties

 Label 26-825e13-1.1-c5e13-0-1 Degree $26$ Conductor $8.202\times 10^{37}$ Sign $1$ Analytic cond. $3.81055\times 10^{27}$ Root an. cond. $11.5028$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 13·2-s + 117·3-s − 19·4-s + 1.52e3·6-s + 304·7-s − 985·8-s + 7.37e3·9-s + 1.57e3·11-s − 2.22e3·12-s + 986·13-s + 3.95e3·14-s − 3.07e3·16-s + 1.47e3·17-s + 9.58e4·18-s + 270·19-s + 3.55e4·21-s + 2.04e4·22-s + 9.08e3·23-s − 1.15e5·24-s + 1.28e4·26-s + 3.31e5·27-s − 5.77e3·28-s + 1.19e4·29-s + 1.90e4·31-s + 2.43e4·32-s + 1.84e5·33-s + 1.91e4·34-s + ⋯
 L(s)  = 1 + 2.29·2-s + 7.50·3-s − 0.593·4-s + 17.2·6-s + 2.34·7-s − 5.44·8-s + 91/3·9-s + 3.91·11-s − 4.45·12-s + 1.61·13-s + 5.38·14-s − 2.99·16-s + 1.23·17-s + 69.7·18-s + 0.171·19-s + 17.5·21-s + 9.00·22-s + 3.58·23-s − 40.8·24-s + 3.71·26-s + 87.5·27-s − 1.39·28-s + 2.63·29-s + 3.56·31-s + 4.20·32-s + 29.4·33-s + 2.84·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 5^{26} \cdot 11^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{13} \cdot 5^{26} \cdot 11^{13}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$26$$ Conductor: $$3^{13} \cdot 5^{26} \cdot 11^{13}$$ Sign: $1$ Analytic conductor: $$3.81055\times 10^{27}$$ Root analytic conductor: $$11.5028$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(26,\ 3^{13} \cdot 5^{26} \cdot 11^{13} ,\ ( \ : [5/2]^{13} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$976253.9603$$ $$L(\frac12)$$ $$\approx$$ $$976253.9603$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$( 1 - p^{2} T )^{13}$$
5 $$1$$
11 $$( 1 - p^{2} T )^{13}$$
good2 $$1 - 13 T + 47 p^{2} T^{2} - 853 p T^{3} + 16015 T^{4} - 119693 T^{5} + 56587 p^{4} T^{6} - 46391 p^{7} T^{7} + 1240073 p^{5} T^{8} - 7379225 p^{5} T^{9} + 5664157 p^{8} T^{10} - 31554979 p^{8} T^{11} + 185606923 p^{8} T^{12} - 1012691925 p^{8} T^{13} + 185606923 p^{13} T^{14} - 31554979 p^{18} T^{15} + 5664157 p^{23} T^{16} - 7379225 p^{25} T^{17} + 1240073 p^{30} T^{18} - 46391 p^{37} T^{19} + 56587 p^{39} T^{20} - 119693 p^{40} T^{21} + 16015 p^{45} T^{22} - 853 p^{51} T^{23} + 47 p^{57} T^{24} - 13 p^{60} T^{25} + p^{65} T^{26}$$
7 $$1 - 304 T + 148175 T^{2} - 5007804 p T^{3} + 10379420630 T^{4} - 2070657148136 T^{5} + 67715580083110 p T^{6} - 82863185149141924 T^{7} + 15911523496088400231 T^{8} -$$$$24\!\cdots\!28$$$$T^{9} +$$$$41\!\cdots\!61$$$$T^{10} -$$$$82\!\cdots\!16$$$$p T^{11} +$$$$86\!\cdots\!92$$$$T^{12} -$$$$10\!\cdots\!68$$$$T^{13} +$$$$86\!\cdots\!92$$$$p^{5} T^{14} -$$$$82\!\cdots\!16$$$$p^{11} T^{15} +$$$$41\!\cdots\!61$$$$p^{15} T^{16} -$$$$24\!\cdots\!28$$$$p^{20} T^{17} + 15911523496088400231 p^{25} T^{18} - 82863185149141924 p^{30} T^{19} + 67715580083110 p^{36} T^{20} - 2070657148136 p^{40} T^{21} + 10379420630 p^{45} T^{22} - 5007804 p^{51} T^{23} + 148175 p^{55} T^{24} - 304 p^{60} T^{25} + p^{65} T^{26}$$
13 $$1 - 986 T + 2671789 T^{2} - 2023899532 T^{3} + 3012394210590 T^{4} - 1765398453965540 T^{5} + 1912928636122212182 T^{6} -$$$$82\!\cdots\!32$$$$T^{7} +$$$$77\!\cdots\!75$$$$T^{8} -$$$$20\!\cdots\!42$$$$T^{9} +$$$$17\!\cdots\!83$$$$p T^{10} -$$$$13\!\cdots\!64$$$$T^{11} +$$$$61\!\cdots\!72$$$$T^{12} +$$$$57\!\cdots\!08$$$$T^{13} +$$$$61\!\cdots\!72$$$$p^{5} T^{14} -$$$$13\!\cdots\!64$$$$p^{10} T^{15} +$$$$17\!\cdots\!83$$$$p^{16} T^{16} -$$$$20\!\cdots\!42$$$$p^{20} T^{17} +$$$$77\!\cdots\!75$$$$p^{25} T^{18} -$$$$82\!\cdots\!32$$$$p^{30} T^{19} + 1912928636122212182 p^{35} T^{20} - 1765398453965540 p^{40} T^{21} + 3012394210590 p^{45} T^{22} - 2023899532 p^{50} T^{23} + 2671789 p^{55} T^{24} - 986 p^{60} T^{25} + p^{65} T^{26}$$
17 $$1 - 1476 T + 7317849 T^{2} - 9821611604 T^{3} + 25436307684050 T^{4} - 31043859533242160 T^{5} + 53783525555433675522 T^{6} -$$$$57\!\cdots\!76$$$$T^{7} +$$$$71\!\cdots\!19$$$$T^{8} -$$$$58\!\cdots\!80$$$$T^{9} +$$$$49\!\cdots\!83$$$$T^{10} -$$$$10\!\cdots\!92$$$$T^{11} -$$$$16\!\cdots\!92$$$$T^{12} +$$$$39\!\cdots\!32$$$$T^{13} -$$$$16\!\cdots\!92$$$$p^{5} T^{14} -$$$$10\!\cdots\!92$$$$p^{10} T^{15} +$$$$49\!\cdots\!83$$$$p^{15} T^{16} -$$$$58\!\cdots\!80$$$$p^{20} T^{17} +$$$$71\!\cdots\!19$$$$p^{25} T^{18} -$$$$57\!\cdots\!76$$$$p^{30} T^{19} + 53783525555433675522 p^{35} T^{20} - 31043859533242160 p^{40} T^{21} + 25436307684050 p^{45} T^{22} - 9821611604 p^{50} T^{23} + 7317849 p^{55} T^{24} - 1476 p^{60} T^{25} + p^{65} T^{26}$$
19 $$1 - 270 T + 758797 p T^{2} + 2426279096 T^{3} + 111917931054446 T^{4} + 2300910280153292 p T^{5} +$$$$62\!\cdots\!34$$$$T^{6} +$$$$32\!\cdots\!20$$$$T^{7} +$$$$26\!\cdots\!31$$$$T^{8} +$$$$15\!\cdots\!30$$$$T^{9} +$$$$50\!\cdots\!07$$$$p T^{10} +$$$$55\!\cdots\!52$$$$T^{11} +$$$$28\!\cdots\!08$$$$T^{12} +$$$$15\!\cdots\!48$$$$T^{13} +$$$$28\!\cdots\!08$$$$p^{5} T^{14} +$$$$55\!\cdots\!52$$$$p^{10} T^{15} +$$$$50\!\cdots\!07$$$$p^{16} T^{16} +$$$$15\!\cdots\!30$$$$p^{20} T^{17} +$$$$26\!\cdots\!31$$$$p^{25} T^{18} +$$$$32\!\cdots\!20$$$$p^{30} T^{19} +$$$$62\!\cdots\!34$$$$p^{35} T^{20} + 2300910280153292 p^{41} T^{21} + 111917931054446 p^{45} T^{22} + 2426279096 p^{50} T^{23} + 758797 p^{56} T^{24} - 270 p^{60} T^{25} + p^{65} T^{26}$$
23 $$1 - 9084 T + 82430095 T^{2} - 509463156256 T^{3} + 2908181073681606 T^{4} - 13950384893685686152 T^{5} +$$$$61\!\cdots\!38$$$$T^{6} -$$$$24\!\cdots\!20$$$$T^{7} +$$$$91\!\cdots\!63$$$$T^{8} -$$$$31\!\cdots\!36$$$$T^{9} +$$$$99\!\cdots\!85$$$$T^{10} -$$$$12\!\cdots\!56$$$$p T^{11} +$$$$82\!\cdots\!16$$$$T^{12} -$$$$21\!\cdots\!28$$$$T^{13} +$$$$82\!\cdots\!16$$$$p^{5} T^{14} -$$$$12\!\cdots\!56$$$$p^{11} T^{15} +$$$$99\!\cdots\!85$$$$p^{15} T^{16} -$$$$31\!\cdots\!36$$$$p^{20} T^{17} +$$$$91\!\cdots\!63$$$$p^{25} T^{18} -$$$$24\!\cdots\!20$$$$p^{30} T^{19} +$$$$61\!\cdots\!38$$$$p^{35} T^{20} - 13950384893685686152 p^{40} T^{21} + 2908181073681606 p^{45} T^{22} - 509463156256 p^{50} T^{23} + 82430095 p^{55} T^{24} - 9084 p^{60} T^{25} + p^{65} T^{26}$$
29 $$1 - 11952 T + 228564481 T^{2} - 2283925450400 T^{3} + 24641310652347350 T^{4} -$$$$20\!\cdots\!68$$$$T^{5} +$$$$16\!\cdots\!66$$$$T^{6} -$$$$11\!\cdots\!40$$$$T^{7} +$$$$27\!\cdots\!63$$$$p T^{8} -$$$$48\!\cdots\!12$$$$T^{9} +$$$$27\!\cdots\!83$$$$T^{10} -$$$$14\!\cdots\!92$$$$T^{11} +$$$$74\!\cdots\!00$$$$T^{12} -$$$$34\!\cdots\!72$$$$T^{13} +$$$$74\!\cdots\!00$$$$p^{5} T^{14} -$$$$14\!\cdots\!92$$$$p^{10} T^{15} +$$$$27\!\cdots\!83$$$$p^{15} T^{16} -$$$$48\!\cdots\!12$$$$p^{20} T^{17} +$$$$27\!\cdots\!63$$$$p^{26} T^{18} -$$$$11\!\cdots\!40$$$$p^{30} T^{19} +$$$$16\!\cdots\!66$$$$p^{35} T^{20} -$$$$20\!\cdots\!68$$$$p^{40} T^{21} + 24641310652347350 p^{45} T^{22} - 2283925450400 p^{50} T^{23} + 228564481 p^{55} T^{24} - 11952 p^{60} T^{25} + p^{65} T^{26}$$
31 $$1 - 616 p T + 407544523 T^{2} - 5100813612176 T^{3} + 64106317678732454 T^{4} -$$$$60\!\cdots\!20$$$$T^{5} +$$$$55\!\cdots\!26$$$$T^{6} -$$$$42\!\cdots\!84$$$$T^{7} +$$$$31\!\cdots\!83$$$$T^{8} -$$$$19\!\cdots\!56$$$$T^{9} +$$$$41\!\cdots\!31$$$$p T^{10} -$$$$71\!\cdots\!72$$$$T^{11} +$$$$42\!\cdots\!88$$$$T^{12} -$$$$21\!\cdots\!64$$$$T^{13} +$$$$42\!\cdots\!88$$$$p^{5} T^{14} -$$$$71\!\cdots\!72$$$$p^{10} T^{15} +$$$$41\!\cdots\!31$$$$p^{16} T^{16} -$$$$19\!\cdots\!56$$$$p^{20} T^{17} +$$$$31\!\cdots\!83$$$$p^{25} T^{18} -$$$$42\!\cdots\!84$$$$p^{30} T^{19} +$$$$55\!\cdots\!26$$$$p^{35} T^{20} -$$$$60\!\cdots\!20$$$$p^{40} T^{21} + 64106317678732454 p^{45} T^{22} - 5100813612176 p^{50} T^{23} + 407544523 p^{55} T^{24} - 616 p^{61} T^{25} + p^{65} T^{26}$$
37 $$1 - 39964 T + 1135425177 T^{2} - 22868960014288 T^{3} + 377171933788233702 T^{4} -$$$$51\!\cdots\!80$$$$T^{5} +$$$$61\!\cdots\!66$$$$T^{6} -$$$$65\!\cdots\!60$$$$T^{7} +$$$$65\!\cdots\!83$$$$T^{8} -$$$$62\!\cdots\!96$$$$T^{9} +$$$$59\!\cdots\!55$$$$T^{10} -$$$$55\!\cdots\!48$$$$T^{11} +$$$$49\!\cdots\!48$$$$T^{12} -$$$$42\!\cdots\!36$$$$T^{13} +$$$$49\!\cdots\!48$$$$p^{5} T^{14} -$$$$55\!\cdots\!48$$$$p^{10} T^{15} +$$$$59\!\cdots\!55$$$$p^{15} T^{16} -$$$$62\!\cdots\!96$$$$p^{20} T^{17} +$$$$65\!\cdots\!83$$$$p^{25} T^{18} -$$$$65\!\cdots\!60$$$$p^{30} T^{19} +$$$$61\!\cdots\!66$$$$p^{35} T^{20} -$$$$51\!\cdots\!80$$$$p^{40} T^{21} + 377171933788233702 p^{45} T^{22} - 22868960014288 p^{50} T^{23} + 1135425177 p^{55} T^{24} - 39964 p^{60} T^{25} + p^{65} T^{26}$$
41 $$1 - 35184 T + 1548060853 T^{2} - 37586637775328 T^{3} + 991257406888164398 T^{4} -$$$$18\!\cdots\!96$$$$T^{5} +$$$$37\!\cdots\!62$$$$T^{6} -$$$$60\!\cdots\!36$$$$T^{7} +$$$$98\!\cdots\!79$$$$T^{8} -$$$$13\!\cdots\!52$$$$T^{9} +$$$$19\!\cdots\!75$$$$T^{10} -$$$$23\!\cdots\!68$$$$T^{11} +$$$$28\!\cdots\!32$$$$T^{12} -$$$$30\!\cdots\!72$$$$T^{13} +$$$$28\!\cdots\!32$$$$p^{5} T^{14} -$$$$23\!\cdots\!68$$$$p^{10} T^{15} +$$$$19\!\cdots\!75$$$$p^{15} T^{16} -$$$$13\!\cdots\!52$$$$p^{20} T^{17} +$$$$98\!\cdots\!79$$$$p^{25} T^{18} -$$$$60\!\cdots\!36$$$$p^{30} T^{19} +$$$$37\!\cdots\!62$$$$p^{35} T^{20} -$$$$18\!\cdots\!96$$$$p^{40} T^{21} + 991257406888164398 p^{45} T^{22} - 37586637775328 p^{50} T^{23} + 1548060853 p^{55} T^{24} - 35184 p^{60} T^{25} + p^{65} T^{26}$$
43 $$1 + 96 T + 889829231 T^{2} + 2468588488300 T^{3} + 375368995071502382 T^{4} +$$$$16\!\cdots\!28$$$$T^{5} +$$$$10\!\cdots\!02$$$$T^{6} +$$$$48\!\cdots\!40$$$$T^{7} +$$$$23\!\cdots\!23$$$$T^{8} +$$$$81\!\cdots\!68$$$$T^{9} +$$$$43\!\cdots\!41$$$$T^{10} +$$$$10\!\cdots\!00$$$$T^{11} +$$$$68\!\cdots\!40$$$$T^{12} +$$$$15\!\cdots\!32$$$$T^{13} +$$$$68\!\cdots\!40$$$$p^{5} T^{14} +$$$$10\!\cdots\!00$$$$p^{10} T^{15} +$$$$43\!\cdots\!41$$$$p^{15} T^{16} +$$$$81\!\cdots\!68$$$$p^{20} T^{17} +$$$$23\!\cdots\!23$$$$p^{25} T^{18} +$$$$48\!\cdots\!40$$$$p^{30} T^{19} +$$$$10\!\cdots\!02$$$$p^{35} T^{20} +$$$$16\!\cdots\!28$$$$p^{40} T^{21} + 375368995071502382 p^{45} T^{22} + 2468588488300 p^{50} T^{23} + 889829231 p^{55} T^{24} + 96 p^{60} T^{25} + p^{65} T^{26}$$
47 $$1 - 34984 T + 2528568823 T^{2} - 75327395182912 T^{3} + 3058881329739106070 T^{4} -$$$$77\!\cdots\!36$$$$T^{5} +$$$$23\!\cdots\!78$$$$T^{6} -$$$$51\!\cdots\!80$$$$T^{7} +$$$$12\!\cdots\!39$$$$T^{8} -$$$$23\!\cdots\!12$$$$T^{9} +$$$$48\!\cdots\!09$$$$T^{10} -$$$$81\!\cdots\!20$$$$T^{11} +$$$$14\!\cdots\!60$$$$T^{12} -$$$$21\!\cdots\!36$$$$T^{13} +$$$$14\!\cdots\!60$$$$p^{5} T^{14} -$$$$81\!\cdots\!20$$$$p^{10} T^{15} +$$$$48\!\cdots\!09$$$$p^{15} T^{16} -$$$$23\!\cdots\!12$$$$p^{20} T^{17} +$$$$12\!\cdots\!39$$$$p^{25} T^{18} -$$$$51\!\cdots\!80$$$$p^{30} T^{19} +$$$$23\!\cdots\!78$$$$p^{35} T^{20} -$$$$77\!\cdots\!36$$$$p^{40} T^{21} + 3058881329739106070 p^{45} T^{22} - 75327395182912 p^{50} T^{23} + 2528568823 p^{55} T^{24} - 34984 p^{60} T^{25} + p^{65} T^{26}$$
53 $$1 - 22984 T + 3071325421 T^{2} - 1066420163840 p T^{3} + 4745257757977091606 T^{4} -$$$$76\!\cdots\!88$$$$T^{5} +$$$$50\!\cdots\!74$$$$T^{6} -$$$$71\!\cdots\!52$$$$T^{7} +$$$$39\!\cdots\!91$$$$T^{8} -$$$$51\!\cdots\!04$$$$T^{9} +$$$$24\!\cdots\!75$$$$T^{10} -$$$$29\!\cdots\!56$$$$T^{11} +$$$$12\!\cdots\!40$$$$T^{12} -$$$$13\!\cdots\!16$$$$T^{13} +$$$$12\!\cdots\!40$$$$p^{5} T^{14} -$$$$29\!\cdots\!56$$$$p^{10} T^{15} +$$$$24\!\cdots\!75$$$$p^{15} T^{16} -$$$$51\!\cdots\!04$$$$p^{20} T^{17} +$$$$39\!\cdots\!91$$$$p^{25} T^{18} -$$$$71\!\cdots\!52$$$$p^{30} T^{19} +$$$$50\!\cdots\!74$$$$p^{35} T^{20} -$$$$76\!\cdots\!88$$$$p^{40} T^{21} + 4745257757977091606 p^{45} T^{22} - 1066420163840 p^{51} T^{23} + 3071325421 p^{55} T^{24} - 22984 p^{60} T^{25} + p^{65} T^{26}$$
59 $$1 + 9192 T + 5651032607 T^{2} + 68244331570480 T^{3} + 16132746466152260142 T^{4} +$$$$22\!\cdots\!72$$$$T^{5} +$$$$30\!\cdots\!70$$$$T^{6} +$$$$44\!\cdots\!20$$$$T^{7} +$$$$43\!\cdots\!19$$$$T^{8} +$$$$62\!\cdots\!64$$$$T^{9} +$$$$48\!\cdots\!33$$$$T^{10} +$$$$65\!\cdots\!00$$$$T^{11} +$$$$42\!\cdots\!48$$$$T^{12} +$$$$52\!\cdots\!44$$$$T^{13} +$$$$42\!\cdots\!48$$$$p^{5} T^{14} +$$$$65\!\cdots\!00$$$$p^{10} T^{15} +$$$$48\!\cdots\!33$$$$p^{15} T^{16} +$$$$62\!\cdots\!64$$$$p^{20} T^{17} +$$$$43\!\cdots\!19$$$$p^{25} T^{18} +$$$$44\!\cdots\!20$$$$p^{30} T^{19} +$$$$30\!\cdots\!70$$$$p^{35} T^{20} +$$$$22\!\cdots\!72$$$$p^{40} T^{21} + 16132746466152260142 p^{45} T^{22} + 68244331570480 p^{50} T^{23} + 5651032607 p^{55} T^{24} + 9192 p^{60} T^{25} + p^{65} T^{26}$$
61 $$1 - 5438 T + 5725645109 T^{2} - 44760935121648 T^{3} + 15911991944583058454 T^{4} -$$$$13\!\cdots\!84$$$$T^{5} +$$$$29\!\cdots\!02$$$$T^{6} -$$$$21\!\cdots\!08$$$$T^{7} +$$$$40\!\cdots\!27$$$$T^{8} -$$$$23\!\cdots\!34$$$$T^{9} +$$$$44\!\cdots\!27$$$$T^{10} -$$$$19\!\cdots\!28$$$$T^{11} +$$$$42\!\cdots\!44$$$$T^{12} -$$$$15\!\cdots\!48$$$$T^{13} +$$$$42\!\cdots\!44$$$$p^{5} T^{14} -$$$$19\!\cdots\!28$$$$p^{10} T^{15} +$$$$44\!\cdots\!27$$$$p^{15} T^{16} -$$$$23\!\cdots\!34$$$$p^{20} T^{17} +$$$$40\!\cdots\!27$$$$p^{25} T^{18} -$$$$21\!\cdots\!08$$$$p^{30} T^{19} +$$$$29\!\cdots\!02$$$$p^{35} T^{20} -$$$$13\!\cdots\!84$$$$p^{40} T^{21} + 15911991944583058454 p^{45} T^{22} - 44760935121648 p^{50} T^{23} + 5725645109 p^{55} T^{24} - 5438 p^{60} T^{25} + p^{65} T^{26}$$
67 $$1 - 71508 T + 10373805919 T^{2} - 486912193719824 T^{3} + 42594134053536929622 T^{4} -$$$$14\!\cdots\!44$$$$T^{5} +$$$$10\!\cdots\!66$$$$T^{6} -$$$$24\!\cdots\!44$$$$T^{7} +$$$$18\!\cdots\!19$$$$T^{8} -$$$$33\!\cdots\!72$$$$T^{9} +$$$$29\!\cdots\!61$$$$T^{10} -$$$$40\!\cdots\!16$$$$T^{11} +$$$$42\!\cdots\!28$$$$T^{12} -$$$$50\!\cdots\!68$$$$T^{13} +$$$$42\!\cdots\!28$$$$p^{5} T^{14} -$$$$40\!\cdots\!16$$$$p^{10} T^{15} +$$$$29\!\cdots\!61$$$$p^{15} T^{16} -$$$$33\!\cdots\!72$$$$p^{20} T^{17} +$$$$18\!\cdots\!19$$$$p^{25} T^{18} -$$$$24\!\cdots\!44$$$$p^{30} T^{19} +$$$$10\!\cdots\!66$$$$p^{35} T^{20} -$$$$14\!\cdots\!44$$$$p^{40} T^{21} + 42594134053536929622 p^{45} T^{22} - 486912193719824 p^{50} T^{23} + 10373805919 p^{55} T^{24} - 71508 p^{60} T^{25} + p^{65} T^{26}$$
71 $$1 - 101700 T + 15607483407 T^{2} - 1070479982775744 T^{3} + 96719658134156434070 T^{4} -$$$$49\!\cdots\!32$$$$T^{5} +$$$$33\!\cdots\!94$$$$T^{6} -$$$$13\!\cdots\!40$$$$T^{7} +$$$$81\!\cdots\!43$$$$T^{8} -$$$$26\!\cdots\!72$$$$T^{9} +$$$$16\!\cdots\!61$$$$T^{10} -$$$$49\!\cdots\!84$$$$T^{11} +$$$$30\!\cdots\!60$$$$T^{12} -$$$$89\!\cdots\!28$$$$T^{13} +$$$$30\!\cdots\!60$$$$p^{5} T^{14} -$$$$49\!\cdots\!84$$$$p^{10} T^{15} +$$$$16\!\cdots\!61$$$$p^{15} T^{16} -$$$$26\!\cdots\!72$$$$p^{20} T^{17} +$$$$81\!\cdots\!43$$$$p^{25} T^{18} -$$$$13\!\cdots\!40$$$$p^{30} T^{19} +$$$$33\!\cdots\!94$$$$p^{35} T^{20} -$$$$49\!\cdots\!32$$$$p^{40} T^{21} + 96719658134156434070 p^{45} T^{22} - 1070479982775744 p^{50} T^{23} + 15607483407 p^{55} T^{24} - 101700 p^{60} T^{25} + p^{65} T^{26}$$
73 $$1 - 77390 T + 13858326365 T^{2} - 752995043500156 T^{3} + 83136418272537378278 T^{4} -$$$$32\!\cdots\!40$$$$T^{5} +$$$$30\!\cdots\!78$$$$T^{6} -$$$$80\!\cdots\!88$$$$T^{7} +$$$$74\!\cdots\!51$$$$T^{8} -$$$$87\!\cdots\!42$$$$T^{9} +$$$$13\!\cdots\!79$$$$T^{10} +$$$$98\!\cdots\!44$$$$T^{11} +$$$$21\!\cdots\!16$$$$T^{12} +$$$$49\!\cdots\!44$$$$T^{13} +$$$$21\!\cdots\!16$$$$p^{5} T^{14} +$$$$98\!\cdots\!44$$$$p^{10} T^{15} +$$$$13\!\cdots\!79$$$$p^{15} T^{16} -$$$$87\!\cdots\!42$$$$p^{20} T^{17} +$$$$74\!\cdots\!51$$$$p^{25} T^{18} -$$$$80\!\cdots\!88$$$$p^{30} T^{19} +$$$$30\!\cdots\!78$$$$p^{35} T^{20} -$$$$32\!\cdots\!40$$$$p^{40} T^{21} + 83136418272537378278 p^{45} T^{22} - 752995043500156 p^{50} T^{23} + 13858326365 p^{55} T^{24} - 77390 p^{60} T^{25} + p^{65} T^{26}$$
79 $$1 - 93954 T + 28347949367 T^{2} - 2720572293016912 T^{3} + 5146005739018291690 p T^{4} -$$$$36\!\cdots\!28$$$$T^{5} +$$$$38\!\cdots\!14$$$$T^{6} -$$$$31\!\cdots\!24$$$$T^{7} +$$$$26\!\cdots\!75$$$$T^{8} -$$$$19\!\cdots\!54$$$$T^{9} +$$$$13\!\cdots\!45$$$$T^{10} -$$$$86\!\cdots\!56$$$$T^{11} +$$$$53\!\cdots\!52$$$$T^{12} -$$$$30\!\cdots\!44$$$$T^{13} +$$$$53\!\cdots\!52$$$$p^{5} T^{14} -$$$$86\!\cdots\!56$$$$p^{10} T^{15} +$$$$13\!\cdots\!45$$$$p^{15} T^{16} -$$$$19\!\cdots\!54$$$$p^{20} T^{17} +$$$$26\!\cdots\!75$$$$p^{25} T^{18} -$$$$31\!\cdots\!24$$$$p^{30} T^{19} +$$$$38\!\cdots\!14$$$$p^{35} T^{20} -$$$$36\!\cdots\!28$$$$p^{40} T^{21} + 5146005739018291690 p^{46} T^{22} - 2720572293016912 p^{50} T^{23} + 28347949367 p^{55} T^{24} - 93954 p^{60} T^{25} + p^{65} T^{26}$$
83 $$1 - 185918 T + 51932524135 T^{2} - 7074572781792356 T^{3} +$$$$11\!\cdots\!78$$$$T^{4} -$$$$12\!\cdots\!80$$$$T^{5} +$$$$15\!\cdots\!66$$$$T^{6} -$$$$13\!\cdots\!08$$$$T^{7} +$$$$13\!\cdots\!83$$$$T^{8} -$$$$10\!\cdots\!70$$$$T^{9} +$$$$86\!\cdots\!17$$$$T^{10} -$$$$59\!\cdots\!12$$$$T^{11} +$$$$43\!\cdots\!48$$$$T^{12} -$$$$26\!\cdots\!80$$$$T^{13} +$$$$43\!\cdots\!48$$$$p^{5} T^{14} -$$$$59\!\cdots\!12$$$$p^{10} T^{15} +$$$$86\!\cdots\!17$$$$p^{15} T^{16} -$$$$10\!\cdots\!70$$$$p^{20} T^{17} +$$$$13\!\cdots\!83$$$$p^{25} T^{18} -$$$$13\!\cdots\!08$$$$p^{30} T^{19} +$$$$15\!\cdots\!66$$$$p^{35} T^{20} -$$$$12\!\cdots\!80$$$$p^{40} T^{21} +$$$$11\!\cdots\!78$$$$p^{45} T^{22} - 7074572781792356 p^{50} T^{23} + 51932524135 p^{55} T^{24} - 185918 p^{60} T^{25} + p^{65} T^{26}$$
89 $$1 + 18418 T + 29814178389 T^{2} + 176697032177160 T^{3} +$$$$47\!\cdots\!30$$$$T^{4} -$$$$12\!\cdots\!32$$$$T^{5} +$$$$54\!\cdots\!62$$$$T^{6} -$$$$44\!\cdots\!80$$$$T^{7} +$$$$50\!\cdots\!67$$$$T^{8} -$$$$57\!\cdots\!66$$$$T^{9} +$$$$38\!\cdots\!87$$$$T^{10} -$$$$49\!\cdots\!92$$$$T^{11} +$$$$24\!\cdots\!08$$$$T^{12} -$$$$31\!\cdots\!16$$$$T^{13} +$$$$24\!\cdots\!08$$$$p^{5} T^{14} -$$$$49\!\cdots\!92$$$$p^{10} T^{15} +$$$$38\!\cdots\!87$$$$p^{15} T^{16} -$$$$57\!\cdots\!66$$$$p^{20} T^{17} +$$$$50\!\cdots\!67$$$$p^{25} T^{18} -$$$$44\!\cdots\!80$$$$p^{30} T^{19} +$$$$54\!\cdots\!62$$$$p^{35} T^{20} -$$$$12\!\cdots\!32$$$$p^{40} T^{21} +$$$$47\!\cdots\!30$$$$p^{45} T^{22} + 176697032177160 p^{50} T^{23} + 29814178389 p^{55} T^{24} + 18418 p^{60} T^{25} + p^{65} T^{26}$$
97 $$1 - 94312 T + 73187679013 T^{2} - 5766958801631264 T^{3} +$$$$25\!\cdots\!46$$$$T^{4} -$$$$17\!\cdots\!36$$$$T^{5} +$$$$58\!\cdots\!34$$$$T^{6} -$$$$34\!\cdots\!44$$$$T^{7} +$$$$98\!\cdots\!67$$$$T^{8} -$$$$51\!\cdots\!88$$$$T^{9} +$$$$12\!\cdots\!55$$$$T^{10} -$$$$60\!\cdots\!36$$$$T^{11} +$$$$13\!\cdots\!08$$$$T^{12} -$$$$57\!\cdots\!64$$$$T^{13} +$$$$13\!\cdots\!08$$$$p^{5} T^{14} -$$$$60\!\cdots\!36$$$$p^{10} T^{15} +$$$$12\!\cdots\!55$$$$p^{15} T^{16} -$$$$51\!\cdots\!88$$$$p^{20} T^{17} +$$$$98\!\cdots\!67$$$$p^{25} T^{18} -$$$$34\!\cdots\!44$$$$p^{30} T^{19} +$$$$58\!\cdots\!34$$$$p^{35} T^{20} -$$$$17\!\cdots\!36$$$$p^{40} T^{21} +$$$$25\!\cdots\!46$$$$p^{45} T^{22} - 5766958801631264 p^{50} T^{23} + 73187679013 p^{55} T^{24} - 94312 p^{60} T^{25} + p^{65} T^{26}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−2.35747004406896197428414827142, −2.31629596320909887117205713364, −2.26418626274941352210762176696, −2.26210101027825062205978438853, −2.02552505856185953283847959589, −1.98683395979185323760915482646, −1.92800345538882350535471115198, −1.70733469467946204101653446186, −1.59460467851108882611484848166, −1.56789029136306637902983738139, −1.56004984908802056305050724872, −1.36460481638896279307194559772, −1.28787034865605385716716951962, −1.17457296588602887316471906788, −1.12630357207077386271308885524, −0.952802378600297660443546801841, −0.918966766439278514611563003644, −0.884452402298638085814471200814, −0.78062203901852461632972930591, −0.70634298584810316347685686403, −0.64924553301945875929470892059, −0.57237827489796841002637860026, −0.51588955211323301310764933248, −0.49791808227505764499553871835, −0.48558607229114015980491567568, 0.48558607229114015980491567568, 0.49791808227505764499553871835, 0.51588955211323301310764933248, 0.57237827489796841002637860026, 0.64924553301945875929470892059, 0.70634298584810316347685686403, 0.78062203901852461632972930591, 0.884452402298638085814471200814, 0.918966766439278514611563003644, 0.952802378600297660443546801841, 1.12630357207077386271308885524, 1.17457296588602887316471906788, 1.28787034865605385716716951962, 1.36460481638896279307194559772, 1.56004984908802056305050724872, 1.56789029136306637902983738139, 1.59460467851108882611484848166, 1.70733469467946204101653446186, 1.92800345538882350535471115198, 1.98683395979185323760915482646, 2.02552505856185953283847959589, 2.26210101027825062205978438853, 2.26418626274941352210762176696, 2.31629596320909887117205713364, 2.35747004406896197428414827142

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

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