# Properties

 Label 40-384e20-1.1-c5e20-0-4 Degree $40$ Conductor $4.860\times 10^{51}$ Sign $1$ Analytic cond. $6.16361\times 10^{35}$ Root an. cond. $7.84776$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + 2·3-s + 2·9-s + 948·11-s + 328·23-s + 2.50e4·25-s + 678·27-s + 1.89e3·33-s − 1.50e4·37-s + 3.66e4·47-s + 1.51e5·49-s + 6.29e4·59-s − 7.32e4·61-s + 656·69-s + 3.48e4·71-s + 5.25e4·73-s + 5.00e4·75-s + 1.52e4·81-s − 2.25e5·83-s + 7.60e3·97-s + 1.89e3·99-s + 1.24e5·107-s + 1.73e5·109-s − 3.01e4·111-s − 1.06e6·121-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 + 0.128·3-s + 0.00823·9-s + 2.36·11-s + 0.129·23-s + 8·25-s + 0.178·27-s + 0.303·33-s − 1.80·37-s + 2.41·47-s + 9.00·49-s + 2.35·59-s − 2.52·61-s + 0.0165·69-s + 0.821·71-s + 1.15·73-s + 1.02·75-s + 0.257·81-s − 3.58·83-s + 0.0820·97-s + 0.0194·99-s + 1.04·107-s + 1.39·109-s − 0.231·111-s − 6.59·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$40$$ Conductor: $$2^{140} \cdot 3^{20}$$ Sign: $1$ Analytic conductor: $$6.16361\times 10^{35}$$ Root analytic conductor: $$7.84776$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$112.0939181$$ $$L(\frac12)$$ $$\approx$$ $$112.0939181$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - 2 T + 2 T^{2} - 226 p T^{3} - 4169 p T^{4} - 163832 p^{2} T^{5} - 92872 p^{2} T^{6} + 2714200 p^{4} T^{7} - 1259962 p^{7} T^{8} + 5104756 p^{8} T^{9} + 18110828 p^{10} T^{10} + 5104756 p^{13} T^{11} - 1259962 p^{17} T^{12} + 2714200 p^{19} T^{13} - 92872 p^{22} T^{14} - 163832 p^{27} T^{15} - 4169 p^{31} T^{16} - 226 p^{36} T^{17} + 2 p^{40} T^{18} - 2 p^{45} T^{19} + p^{50} T^{20}$$
good5 $$1 - 8 p^{5} T^{2} + 64922862 p T^{4} - 581176159496 p T^{6} + 805161162779173 p^{2} T^{8} -$$$$11\!\cdots\!08$$$$T^{10} +$$$$44\!\cdots\!92$$$$p^{3} T^{12} -$$$$48\!\cdots\!28$$$$p T^{14} +$$$$18\!\cdots\!14$$$$p T^{16} -$$$$13\!\cdots\!24$$$$p^{2} T^{18} +$$$$10\!\cdots\!16$$$$T^{20} -$$$$13\!\cdots\!24$$$$p^{12} T^{22} +$$$$18\!\cdots\!14$$$$p^{21} T^{24} -$$$$48\!\cdots\!28$$$$p^{31} T^{26} +$$$$44\!\cdots\!92$$$$p^{43} T^{28} -$$$$11\!\cdots\!08$$$$p^{50} T^{30} + 805161162779173 p^{62} T^{32} - 581176159496 p^{71} T^{34} + 64922862 p^{81} T^{36} - 8 p^{95} T^{38} + p^{100} T^{40}$$
7 $$1 - 151376 T^{2} + 11636394822 T^{4} - 602773429074896 T^{6} + 23724202393315072525 T^{8} -$$$$76\!\cdots\!68$$$$T^{10} +$$$$20\!\cdots\!20$$$$T^{12} -$$$$50\!\cdots\!84$$$$T^{14} +$$$$15\!\cdots\!74$$$$p T^{16} -$$$$21\!\cdots\!52$$$$T^{18} +$$$$37\!\cdots\!28$$$$T^{20} -$$$$21\!\cdots\!52$$$$p^{10} T^{22} +$$$$15\!\cdots\!74$$$$p^{21} T^{24} -$$$$50\!\cdots\!84$$$$p^{30} T^{26} +$$$$20\!\cdots\!20$$$$p^{40} T^{28} -$$$$76\!\cdots\!68$$$$p^{50} T^{30} + 23724202393315072525 p^{60} T^{32} - 602773429074896 p^{70} T^{34} + 11636394822 p^{80} T^{36} - 151376 p^{90} T^{38} + p^{100} T^{40}$$
11 $$( 1 - 474 T + 867706 T^{2} - 318904062 T^{3} + 2808086013 p^{2} T^{4} - 103919584780056 T^{5} + 85745787142006680 T^{6} - 23781595566239469960 T^{7} +$$$$16\!\cdots\!22$$$$T^{8} -$$$$44\!\cdots\!48$$$$T^{9} +$$$$29\!\cdots\!36$$$$T^{10} -$$$$44\!\cdots\!48$$$$p^{5} T^{11} +$$$$16\!\cdots\!22$$$$p^{10} T^{12} - 23781595566239469960 p^{15} T^{13} + 85745787142006680 p^{20} T^{14} - 103919584780056 p^{25} T^{15} + 2808086013 p^{32} T^{16} - 318904062 p^{35} T^{17} + 867706 p^{40} T^{18} - 474 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
13 $$( 1 + 1727358 T^{2} - 389121024 T^{3} + 1484851902293 T^{4} - 633595156478976 T^{5} + 897326840696086824 T^{6} -$$$$51\!\cdots\!00$$$$T^{7} +$$$$43\!\cdots\!66$$$$T^{8} -$$$$27\!\cdots\!56$$$$T^{9} +$$$$17\!\cdots\!76$$$$T^{10} -$$$$27\!\cdots\!56$$$$p^{5} T^{11} +$$$$43\!\cdots\!66$$$$p^{10} T^{12} -$$$$51\!\cdots\!00$$$$p^{15} T^{13} + 897326840696086824 p^{20} T^{14} - 633595156478976 p^{25} T^{15} + 1484851902293 p^{30} T^{16} - 389121024 p^{35} T^{17} + 1727358 p^{40} T^{18} + p^{50} T^{20} )^{2}$$
17 $$1 - 12402532 T^{2} + 4475762806206 p T^{4} -$$$$31\!\cdots\!88$$$$T^{6} +$$$$97\!\cdots\!17$$$$T^{8} -$$$$24\!\cdots\!20$$$$T^{10} +$$$$54\!\cdots\!88$$$$T^{12} -$$$$10\!\cdots\!08$$$$T^{14} +$$$$18\!\cdots\!38$$$$T^{16} -$$$$30\!\cdots\!72$$$$T^{18} +$$$$44\!\cdots\!08$$$$T^{20} -$$$$30\!\cdots\!72$$$$p^{10} T^{22} +$$$$18\!\cdots\!38$$$$p^{20} T^{24} -$$$$10\!\cdots\!08$$$$p^{30} T^{26} +$$$$54\!\cdots\!88$$$$p^{40} T^{28} -$$$$24\!\cdots\!20$$$$p^{50} T^{30} +$$$$97\!\cdots\!17$$$$p^{60} T^{32} -$$$$31\!\cdots\!88$$$$p^{70} T^{34} + 4475762806206 p^{81} T^{36} - 12402532 p^{90} T^{38} + p^{100} T^{40}$$
19 $$1 - 1217616 p T^{2} + 283813760957590 T^{4} -$$$$24\!\cdots\!04$$$$T^{6} +$$$$15\!\cdots\!25$$$$T^{8} -$$$$82\!\cdots\!56$$$$T^{10} +$$$$36\!\cdots\!96$$$$T^{12} -$$$$14\!\cdots\!24$$$$T^{14} +$$$$24\!\cdots\!14$$$$p T^{16} -$$$$13\!\cdots\!44$$$$T^{18} +$$$$10\!\cdots\!80$$$$p^{2} T^{20} -$$$$13\!\cdots\!44$$$$p^{10} T^{22} +$$$$24\!\cdots\!14$$$$p^{21} T^{24} -$$$$14\!\cdots\!24$$$$p^{30} T^{26} +$$$$36\!\cdots\!96$$$$p^{40} T^{28} -$$$$82\!\cdots\!56$$$$p^{50} T^{30} +$$$$15\!\cdots\!25$$$$p^{60} T^{32} -$$$$24\!\cdots\!04$$$$p^{70} T^{34} + 283813760957590 p^{80} T^{36} - 1217616 p^{91} T^{38} + p^{100} T^{40}$$
23 $$( 1 - 164 T + 31861174 T^{2} - 8842821788 T^{3} + 496564216557245 T^{4} - 254302378049959056 T^{5} +$$$$52\!\cdots\!24$$$$T^{6} -$$$$39\!\cdots\!36$$$$T^{7} +$$$$44\!\cdots\!66$$$$T^{8} -$$$$38\!\cdots\!84$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} -$$$$38\!\cdots\!84$$$$p^{5} T^{11} +$$$$44\!\cdots\!66$$$$p^{10} T^{12} -$$$$39\!\cdots\!36$$$$p^{15} T^{13} +$$$$52\!\cdots\!24$$$$p^{20} T^{14} - 254302378049959056 p^{25} T^{15} + 496564216557245 p^{30} T^{16} - 8842821788 p^{35} T^{17} + 31861174 p^{40} T^{18} - 164 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
29 $$1 - 191113384 T^{2} + 18110181894055494 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$55\!\cdots\!05$$$$T^{8} -$$$$22\!\cdots\!64$$$$T^{10} +$$$$76\!\cdots\!68$$$$T^{12} -$$$$23\!\cdots\!04$$$$T^{14} +$$$$62\!\cdots\!14$$$$T^{16} -$$$$14\!\cdots\!84$$$$T^{18} +$$$$32\!\cdots\!16$$$$T^{20} -$$$$14\!\cdots\!84$$$$p^{10} T^{22} +$$$$62\!\cdots\!14$$$$p^{20} T^{24} -$$$$23\!\cdots\!04$$$$p^{30} T^{26} +$$$$76\!\cdots\!68$$$$p^{40} T^{28} -$$$$22\!\cdots\!64$$$$p^{50} T^{30} +$$$$55\!\cdots\!05$$$$p^{60} T^{32} -$$$$11\!\cdots\!24$$$$p^{70} T^{34} + 18110181894055494 p^{80} T^{36} - 191113384 p^{90} T^{38} + p^{100} T^{40}$$
31 $$1 - 252335328 T^{2} + 33282623187817222 T^{4} -$$$$30\!\cdots\!28$$$$T^{6} +$$$$21\!\cdots\!37$$$$T^{8} -$$$$12\!\cdots\!56$$$$T^{10} +$$$$63\!\cdots\!56$$$$T^{12} -$$$$27\!\cdots\!84$$$$T^{14} +$$$$10\!\cdots\!66$$$$T^{16} -$$$$36\!\cdots\!04$$$$T^{18} +$$$$10\!\cdots\!36$$$$T^{20} -$$$$36\!\cdots\!04$$$$p^{10} T^{22} +$$$$10\!\cdots\!66$$$$p^{20} T^{24} -$$$$27\!\cdots\!84$$$$p^{30} T^{26} +$$$$63\!\cdots\!56$$$$p^{40} T^{28} -$$$$12\!\cdots\!56$$$$p^{50} T^{30} +$$$$21\!\cdots\!37$$$$p^{60} T^{32} -$$$$30\!\cdots\!28$$$$p^{70} T^{34} + 33282623187817222 p^{80} T^{36} - 252335328 p^{90} T^{38} + p^{100} T^{40}$$
37 $$( 1 + 7528 T + 415621758 T^{2} + 2534251060552 T^{3} + 86513533821089413 T^{4} +$$$$44\!\cdots\!08$$$$T^{5} +$$$$11\!\cdots\!68$$$$T^{6} +$$$$52\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!22$$$$T^{8} +$$$$46\!\cdots\!60$$$$T^{9} +$$$$94\!\cdots\!20$$$$T^{10} +$$$$46\!\cdots\!60$$$$p^{5} T^{11} +$$$$12\!\cdots\!22$$$$p^{10} T^{12} +$$$$52\!\cdots\!20$$$$p^{15} T^{13} +$$$$11\!\cdots\!68$$$$p^{20} T^{14} +$$$$44\!\cdots\!08$$$$p^{25} T^{15} + 86513533821089413 p^{30} T^{16} + 2534251060552 p^{35} T^{17} + 415621758 p^{40} T^{18} + 7528 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
41 $$1 - 992892100 T^{2} + 507553214877646398 T^{4} -$$$$17\!\cdots\!64$$$$T^{6} +$$$$48\!\cdots\!93$$$$T^{8} -$$$$10\!\cdots\!80$$$$T^{10} +$$$$20\!\cdots\!48$$$$T^{12} -$$$$33\!\cdots\!08$$$$T^{14} +$$$$49\!\cdots\!10$$$$T^{16} -$$$$65\!\cdots\!64$$$$T^{18} +$$$$78\!\cdots\!68$$$$T^{20} -$$$$65\!\cdots\!64$$$$p^{10} T^{22} +$$$$49\!\cdots\!10$$$$p^{20} T^{24} -$$$$33\!\cdots\!08$$$$p^{30} T^{26} +$$$$20\!\cdots\!48$$$$p^{40} T^{28} -$$$$10\!\cdots\!80$$$$p^{50} T^{30} +$$$$48\!\cdots\!93$$$$p^{60} T^{32} -$$$$17\!\cdots\!64$$$$p^{70} T^{34} + 507553214877646398 p^{80} T^{36} - 992892100 p^{90} T^{38} + p^{100} T^{40}$$
43 $$1 - 1462691120 T^{2} + 1076203296774082038 T^{4} -$$$$52\!\cdots\!20$$$$T^{6} +$$$$19\!\cdots\!73$$$$T^{8} -$$$$56\!\cdots\!40$$$$T^{10} +$$$$14\!\cdots\!92$$$$T^{12} -$$$$30\!\cdots\!84$$$$T^{14} +$$$$57\!\cdots\!70$$$$T^{16} -$$$$99\!\cdots\!88$$$$T^{18} +$$$$15\!\cdots\!12$$$$T^{20} -$$$$99\!\cdots\!88$$$$p^{10} T^{22} +$$$$57\!\cdots\!70$$$$p^{20} T^{24} -$$$$30\!\cdots\!84$$$$p^{30} T^{26} +$$$$14\!\cdots\!92$$$$p^{40} T^{28} -$$$$56\!\cdots\!40$$$$p^{50} T^{30} +$$$$19\!\cdots\!73$$$$p^{60} T^{32} -$$$$52\!\cdots\!20$$$$p^{70} T^{34} + 1076203296774082038 p^{80} T^{36} - 1462691120 p^{90} T^{38} + p^{100} T^{40}$$
47 $$( 1 - 18320 T + 1471207574 T^{2} - 19754962156272 T^{3} + 989833625114591949 T^{4} -$$$$10\!\cdots\!92$$$$T^{5} +$$$$43\!\cdots\!60$$$$T^{6} -$$$$38\!\cdots\!40$$$$T^{7} +$$$$14\!\cdots\!78$$$$T^{8} -$$$$11\!\cdots\!12$$$$T^{9} +$$$$37\!\cdots\!12$$$$T^{10} -$$$$11\!\cdots\!12$$$$p^{5} T^{11} +$$$$14\!\cdots\!78$$$$p^{10} T^{12} -$$$$38\!\cdots\!40$$$$p^{15} T^{13} +$$$$43\!\cdots\!60$$$$p^{20} T^{14} -$$$$10\!\cdots\!92$$$$p^{25} T^{15} + 989833625114591949 p^{30} T^{16} - 19754962156272 p^{35} T^{17} + 1471207574 p^{40} T^{18} - 18320 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
53 $$1 - 4029665128 T^{2} + 7931455217506153830 T^{4} -$$$$10\!\cdots\!40$$$$T^{6} +$$$$98\!\cdots\!97$$$$T^{8} -$$$$75\!\cdots\!84$$$$T^{10} +$$$$48\!\cdots\!36$$$$T^{12} -$$$$27\!\cdots\!56$$$$T^{14} +$$$$13\!\cdots\!38$$$$T^{16} -$$$$64\!\cdots\!60$$$$T^{18} +$$$$27\!\cdots\!76$$$$T^{20} -$$$$64\!\cdots\!60$$$$p^{10} T^{22} +$$$$13\!\cdots\!38$$$$p^{20} T^{24} -$$$$27\!\cdots\!56$$$$p^{30} T^{26} +$$$$48\!\cdots\!36$$$$p^{40} T^{28} -$$$$75\!\cdots\!84$$$$p^{50} T^{30} +$$$$98\!\cdots\!97$$$$p^{60} T^{32} -$$$$10\!\cdots\!40$$$$p^{70} T^{34} + 7931455217506153830 p^{80} T^{36} - 4029665128 p^{90} T^{38} + p^{100} T^{40}$$
59 $$( 1 - 31454 T + 4623295858 T^{2} - 124799446446410 T^{3} + 10014833728553042357 T^{4} -$$$$23\!\cdots\!44$$$$T^{5} +$$$$13\!\cdots\!84$$$$T^{6} -$$$$29\!\cdots\!72$$$$T^{7} +$$$$14\!\cdots\!74$$$$T^{8} -$$$$26\!\cdots\!00$$$$T^{9} +$$$$11\!\cdots\!16$$$$T^{10} -$$$$26\!\cdots\!00$$$$p^{5} T^{11} +$$$$14\!\cdots\!74$$$$p^{10} T^{12} -$$$$29\!\cdots\!72$$$$p^{15} T^{13} +$$$$13\!\cdots\!84$$$$p^{20} T^{14} -$$$$23\!\cdots\!44$$$$p^{25} T^{15} + 10014833728553042357 p^{30} T^{16} - 124799446446410 p^{35} T^{17} + 4623295858 p^{40} T^{18} - 31454 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
61 $$( 1 + 36632 T + 5042515374 T^{2} + 191771260465784 T^{3} + 13110617571305596597 T^{4} +$$$$47\!\cdots\!08$$$$T^{5} +$$$$22\!\cdots\!44$$$$T^{6} +$$$$75\!\cdots\!28$$$$T^{7} +$$$$29\!\cdots\!38$$$$T^{8} +$$$$85\!\cdots\!84$$$$T^{9} +$$$$28\!\cdots\!44$$$$T^{10} +$$$$85\!\cdots\!84$$$$p^{5} T^{11} +$$$$29\!\cdots\!38$$$$p^{10} T^{12} +$$$$75\!\cdots\!28$$$$p^{15} T^{13} +$$$$22\!\cdots\!44$$$$p^{20} T^{14} +$$$$47\!\cdots\!08$$$$p^{25} T^{15} + 13110617571305596597 p^{30} T^{16} + 191771260465784 p^{35} T^{17} + 5042515374 p^{40} T^{18} + 36632 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
67 $$1 - 10843590944 T^{2} + 59375557604770839670 T^{4} -$$$$22\!\cdots\!48$$$$T^{6} +$$$$66\!\cdots\!21$$$$T^{8} -$$$$16\!\cdots\!72$$$$T^{10} +$$$$35\!\cdots\!84$$$$T^{12} -$$$$67\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!50$$$$T^{16} -$$$$17\!\cdots\!24$$$$T^{18} +$$$$25\!\cdots\!48$$$$T^{20} -$$$$17\!\cdots\!24$$$$p^{10} T^{22} +$$$$11\!\cdots\!50$$$$p^{20} T^{24} -$$$$67\!\cdots\!28$$$$p^{30} T^{26} +$$$$35\!\cdots\!84$$$$p^{40} T^{28} -$$$$16\!\cdots\!72$$$$p^{50} T^{30} +$$$$66\!\cdots\!21$$$$p^{60} T^{32} -$$$$22\!\cdots\!48$$$$p^{70} T^{34} + 59375557604770839670 p^{80} T^{36} - 10843590944 p^{90} T^{38} + p^{100} T^{40}$$
71 $$( 1 - 17444 T + 8202464246 T^{2} - 193853763963612 T^{3} + 37489454553586147677 T^{4} -$$$$91\!\cdots\!60$$$$T^{5} +$$$$12\!\cdots\!88$$$$T^{6} -$$$$28\!\cdots\!20$$$$T^{7} +$$$$30\!\cdots\!02$$$$T^{8} -$$$$65\!\cdots\!56$$$$T^{9} +$$$$60\!\cdots\!08$$$$T^{10} -$$$$65\!\cdots\!56$$$$p^{5} T^{11} +$$$$30\!\cdots\!02$$$$p^{10} T^{12} -$$$$28\!\cdots\!20$$$$p^{15} T^{13} +$$$$12\!\cdots\!88$$$$p^{20} T^{14} -$$$$91\!\cdots\!60$$$$p^{25} T^{15} + 37489454553586147677 p^{30} T^{16} - 193853763963612 p^{35} T^{17} + 8202464246 p^{40} T^{18} - 17444 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
73 $$( 1 - 26284 T + 12244859470 T^{2} - 181128130380588 T^{3} + 73557964980644644029 T^{4} -$$$$63\!\cdots\!36$$$$T^{5} +$$$$30\!\cdots\!76$$$$T^{6} -$$$$17\!\cdots\!12$$$$T^{7} +$$$$92\!\cdots\!30$$$$T^{8} -$$$$38\!\cdots\!04$$$$T^{9} +$$$$21\!\cdots\!44$$$$T^{10} -$$$$38\!\cdots\!04$$$$p^{5} T^{11} +$$$$92\!\cdots\!30$$$$p^{10} T^{12} -$$$$17\!\cdots\!12$$$$p^{15} T^{13} +$$$$30\!\cdots\!76$$$$p^{20} T^{14} -$$$$63\!\cdots\!36$$$$p^{25} T^{15} + 73557964980644644029 p^{30} T^{16} - 181128130380588 p^{35} T^{17} + 12244859470 p^{40} T^{18} - 26284 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
79 $$1 - 24163428736 T^{2} +$$$$30\!\cdots\!58$$$$T^{4} -$$$$26\!\cdots\!72$$$$T^{6} +$$$$17\!\cdots\!37$$$$T^{8} -$$$$99\!\cdots\!00$$$$T^{10} +$$$$47\!\cdots\!76$$$$T^{12} -$$$$20\!\cdots\!04$$$$T^{14} +$$$$77\!\cdots\!26$$$$T^{16} -$$$$27\!\cdots\!44$$$$T^{18} +$$$$87\!\cdots\!76$$$$T^{20} -$$$$27\!\cdots\!44$$$$p^{10} T^{22} +$$$$77\!\cdots\!26$$$$p^{20} T^{24} -$$$$20\!\cdots\!04$$$$p^{30} T^{26} +$$$$47\!\cdots\!76$$$$p^{40} T^{28} -$$$$99\!\cdots\!00$$$$p^{50} T^{30} +$$$$17\!\cdots\!37$$$$p^{60} T^{32} -$$$$26\!\cdots\!72$$$$p^{70} T^{34} +$$$$30\!\cdots\!58$$$$p^{80} T^{36} - 24163428736 p^{90} T^{38} + p^{100} T^{40}$$
83 $$( 1 + 112586 T + 19402311130 T^{2} + 1703864185686878 T^{3} +$$$$16\!\cdots\!21$$$$T^{4} +$$$$12\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!48$$$$T^{6} +$$$$70\!\cdots\!32$$$$T^{7} +$$$$55\!\cdots\!30$$$$T^{8} +$$$$35\!\cdots\!76$$$$T^{9} +$$$$24\!\cdots\!96$$$$T^{10} +$$$$35\!\cdots\!76$$$$p^{5} T^{11} +$$$$55\!\cdots\!30$$$$p^{10} T^{12} +$$$$70\!\cdots\!32$$$$p^{15} T^{13} +$$$$10\!\cdots\!48$$$$p^{20} T^{14} +$$$$12\!\cdots\!00$$$$p^{25} T^{15} +$$$$16\!\cdots\!21$$$$p^{30} T^{16} + 1703864185686878 p^{35} T^{17} + 19402311130 p^{40} T^{18} + 112586 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
89 $$1 - 53305266132 T^{2} +$$$$13\!\cdots\!30$$$$T^{4} -$$$$22\!\cdots\!96$$$$T^{6} +$$$$27\!\cdots\!89$$$$T^{8} -$$$$27\!\cdots\!36$$$$T^{10} +$$$$22\!\cdots\!48$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$11\!\cdots\!22$$$$T^{16} -$$$$70\!\cdots\!80$$$$T^{18} +$$$$40\!\cdots\!20$$$$T^{20} -$$$$70\!\cdots\!80$$$$p^{10} T^{22} +$$$$11\!\cdots\!22$$$$p^{20} T^{24} -$$$$16\!\cdots\!56$$$$p^{30} T^{26} +$$$$22\!\cdots\!48$$$$p^{40} T^{28} -$$$$27\!\cdots\!36$$$$p^{50} T^{30} +$$$$27\!\cdots\!89$$$$p^{60} T^{32} -$$$$22\!\cdots\!96$$$$p^{70} T^{34} +$$$$13\!\cdots\!30$$$$p^{80} T^{36} - 53305266132 p^{90} T^{38} + p^{100} T^{40}$$
97 $$( 1 - 3800 T + 22443771702 T^{2} - 608378862672152 T^{3} +$$$$32\!\cdots\!41$$$$T^{4} -$$$$82\!\cdots\!72$$$$T^{5} +$$$$43\!\cdots\!12$$$$T^{6} -$$$$80\!\cdots\!80$$$$T^{7} +$$$$47\!\cdots\!18$$$$T^{8} -$$$$11\!\cdots\!16$$$$T^{9} +$$$$43\!\cdots\!80$$$$T^{10} -$$$$11\!\cdots\!16$$$$p^{5} T^{11} +$$$$47\!\cdots\!18$$$$p^{10} T^{12} -$$$$80\!\cdots\!80$$$$p^{15} T^{13} +$$$$43\!\cdots\!12$$$$p^{20} T^{14} -$$$$82\!\cdots\!72$$$$p^{25} T^{15} +$$$$32\!\cdots\!41$$$$p^{30} T^{16} - 608378862672152 p^{35} T^{17} + 22443771702 p^{40} T^{18} - 3800 p^{45} T^{19} + p^{50} T^{20} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−1.80066944714690752956909443974, −1.72867538717502496119276597304, −1.58435350228500316912635826926, −1.54550054656975866130083785592, −1.52059766969120268016990812974, −1.46389045330974959423673306162, −1.38430794657640139328140294966, −1.30228080348226470675290892320, −1.10671507956286573078553947358, −1.02808718949820091952996802238, −1.01450713240615435649824850453, −0.998720482230269350502981718102, −0.985296476621003394371106370965, −0.935250653704503542229163105988, −0.885619997055228325429045128925, −0.78213996434004788758617978899, −0.73630190093519450803123198803, −0.63676272585343401916022273881, −0.56295526999231804046149506867, −0.50831926072943220782353335932, −0.37206896844195966584935148081, −0.29643591884794886895456736761, −0.26943015696211501423976511159, −0.12799995467724763621122280744, −0.091302754120792168342670812937, 0.091302754120792168342670812937, 0.12799995467724763621122280744, 0.26943015696211501423976511159, 0.29643591884794886895456736761, 0.37206896844195966584935148081, 0.50831926072943220782353335932, 0.56295526999231804046149506867, 0.63676272585343401916022273881, 0.73630190093519450803123198803, 0.78213996434004788758617978899, 0.885619997055228325429045128925, 0.935250653704503542229163105988, 0.985296476621003394371106370965, 0.998720482230269350502981718102, 1.01450713240615435649824850453, 1.02808718949820091952996802238, 1.10671507956286573078553947358, 1.30228080348226470675290892320, 1.38430794657640139328140294966, 1.46389045330974959423673306162, 1.52059766969120268016990812974, 1.54550054656975866130083785592, 1.58435350228500316912635826926, 1.72867538717502496119276597304, 1.80066944714690752956909443974

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

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