# Properties

 Label 40-177e20-1.1-c2e20-0-0 Degree $40$ Conductor $9.109\times 10^{44}$ Sign $1$ Analytic cond. $4.63624\times 10^{13}$ Root an. cond. $2.19611$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 20·4-s − 8·7-s + 30·9-s + 158·16-s + 16·17-s − 60·19-s − 200·25-s − 160·28-s − 60·29-s + 600·36-s + 28·41-s − 316·49-s − 8·53-s − 152·59-s − 240·63-s + 522·64-s + 320·68-s + 92·71-s − 1.20e3·76-s − 420·79-s + 495·81-s − 4.00e3·100-s − 620·107-s − 1.26e3·112-s − 1.20e3·116-s − 128·119-s + 502·121-s + ⋯
 L(s)  = 1 + 5·4-s − 8/7·7-s + 10/3·9-s + 79/8·16-s + 0.941·17-s − 3.15·19-s − 8·25-s − 5.71·28-s − 2.06·29-s + 50/3·36-s + 0.682·41-s − 6.44·49-s − 0.150·53-s − 2.57·59-s − 3.80·63-s + 8.15·64-s + 4.70·68-s + 1.29·71-s − 15.7·76-s − 5.31·79-s + 55/9·81-s − 40·100-s − 5.79·107-s − 11.2·112-s − 10.3·116-s − 1.07·119-s + 4.14·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$40$$ Conductor: $$3^{20} \cdot 59^{20}$$ Sign: $1$ Analytic conductor: $$4.63624\times 10^{13}$$ Root analytic conductor: $$2.19611$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{177} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(40,\ 3^{20} \cdot 59^{20} ,\ ( \ : [1]^{20} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.3449384622$$ $$L(\frac12)$$ $$\approx$$ $$0.3449384622$$ $$L(2)$$ 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 T^{2} )^{10}$$
59 $$1 + 152 T + 18302 T^{2} + 1459304 T^{3} + 109357965 T^{4} + 7084877840 T^{5} + 8624620440 p T^{6} + 9440238320 p^{2} T^{7} + 10696468086 p^{3} T^{8} + 10413087680 p^{4} T^{9} + 10884724444 p^{5} T^{10} + 10413087680 p^{6} T^{11} + 10696468086 p^{7} T^{12} + 9440238320 p^{8} T^{13} + 8624620440 p^{9} T^{14} + 7084877840 p^{10} T^{15} + 109357965 p^{12} T^{16} + 1459304 p^{14} T^{17} + 18302 p^{16} T^{18} + 152 p^{18} T^{19} + p^{20} T^{20}$$
good2 $$1 - 5 p^{2} T^{2} + 121 p T^{4} - 1101 p T^{6} + 16697 T^{8} - 27359 p^{2} T^{10} + 638337 T^{12} - 1686389 p T^{14} + 8176355 p T^{16} - 18292811 p^{2} T^{18} + 303708557 T^{20} - 18292811 p^{6} T^{22} + 8176355 p^{9} T^{24} - 1686389 p^{13} T^{26} + 638337 p^{16} T^{28} - 27359 p^{22} T^{30} + 16697 p^{24} T^{32} - 1101 p^{29} T^{34} + 121 p^{33} T^{36} - 5 p^{38} T^{38} + p^{40} T^{40}$$
5 $$( 1 + 4 p^{2} T^{2} + 54 T^{3} + 4784 T^{4} + 2048 T^{5} + 153714 T^{6} - 24872 T^{7} + 3760957 T^{8} - 2452838 T^{9} + 86828968 T^{10} - 2452838 p^{2} T^{11} + 3760957 p^{4} T^{12} - 24872 p^{6} T^{13} + 153714 p^{8} T^{14} + 2048 p^{10} T^{15} + 4784 p^{12} T^{16} + 54 p^{14} T^{17} + 4 p^{18} T^{18} + p^{20} T^{20} )^{2}$$
7 $$( 1 + 4 T + 26 p T^{2} + 706 T^{3} + 20301 T^{4} + 64858 T^{5} + 1627341 T^{6} + 4395154 T^{7} + 103092678 T^{8} + 242700046 T^{9} + 5496972338 T^{10} + 242700046 p^{2} T^{11} + 103092678 p^{4} T^{12} + 4395154 p^{6} T^{13} + 1627341 p^{8} T^{14} + 64858 p^{10} T^{15} + 20301 p^{12} T^{16} + 706 p^{14} T^{17} + 26 p^{17} T^{18} + 4 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
11 $$1 - 502 T^{2} + 182661 T^{4} - 49609034 T^{6} + 1019503908 p T^{8} - 2179648042490 T^{10} + 375092432998995 T^{12} - 58362606555507626 T^{14} + 8338624859801251671 T^{16} -$$$$11\!\cdots\!12$$$$T^{18} +$$$$13\!\cdots\!80$$$$T^{20} -$$$$11\!\cdots\!12$$$$p^{4} T^{22} + 8338624859801251671 p^{8} T^{24} - 58362606555507626 p^{12} T^{26} + 375092432998995 p^{16} T^{28} - 2179648042490 p^{20} T^{30} + 1019503908 p^{25} T^{32} - 49609034 p^{28} T^{34} + 182661 p^{32} T^{36} - 502 p^{36} T^{38} + p^{40} T^{40}$$
13 $$1 - 1350 T^{2} + 890761 T^{4} - 381297438 T^{6} + 705860804 p^{2} T^{8} - 29555113366886 T^{10} + 6287781440837163 T^{12} - 1253627755704023602 T^{14} +$$$$24\!\cdots\!23$$$$T^{16} -$$$$47\!\cdots\!72$$$$T^{18} +$$$$49\!\cdots\!88$$$$p^{2} T^{20} -$$$$47\!\cdots\!72$$$$p^{4} T^{22} +$$$$24\!\cdots\!23$$$$p^{8} T^{24} - 1253627755704023602 p^{12} T^{26} + 6287781440837163 p^{16} T^{28} - 29555113366886 p^{20} T^{30} + 705860804 p^{26} T^{32} - 381297438 p^{28} T^{34} + 890761 p^{32} T^{36} - 1350 p^{36} T^{38} + p^{40} T^{40}$$
17 $$( 1 - 8 T + 1816 T^{2} - 6228 T^{3} + 1466195 T^{4} + 2814502 T^{5} + 701432321 T^{6} + 5681420338 T^{7} + 236215639120 T^{8} + 3200569946500 T^{9} + 68840644845798 T^{10} + 3200569946500 p^{2} T^{11} + 236215639120 p^{4} T^{12} + 5681420338 p^{6} T^{13} + 701432321 p^{8} T^{14} + 2814502 p^{10} T^{15} + 1466195 p^{12} T^{16} - 6228 p^{14} T^{17} + 1816 p^{16} T^{18} - 8 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
19 $$( 1 + 30 T + 2097 T^{2} + 58432 T^{3} + 2236518 T^{4} + 56806098 T^{5} + 1552758742 T^{6} + 36233869870 T^{7} + 41760314703 p T^{8} + 16961494042914 T^{9} + 318898249393354 T^{10} + 16961494042914 p^{2} T^{11} + 41760314703 p^{5} T^{12} + 36233869870 p^{6} T^{13} + 1552758742 p^{8} T^{14} + 56806098 p^{10} T^{15} + 2236518 p^{12} T^{16} + 58432 p^{14} T^{17} + 2097 p^{16} T^{18} + 30 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
23 $$1 - 5958 T^{2} + 18026833 T^{4} - 36663484032 T^{6} + 56127223009238 T^{8} - 68701771540253066 T^{10} + 69739236701940349758 T^{12} -$$$$60\!\cdots\!86$$$$T^{14} +$$$$44\!\cdots\!21$$$$T^{16} -$$$$28\!\cdots\!50$$$$T^{18} +$$$$16\!\cdots\!98$$$$T^{20} -$$$$28\!\cdots\!50$$$$p^{4} T^{22} +$$$$44\!\cdots\!21$$$$p^{8} T^{24} -$$$$60\!\cdots\!86$$$$p^{12} T^{26} + 69739236701940349758 p^{16} T^{28} - 68701771540253066 p^{20} T^{30} + 56127223009238 p^{24} T^{32} - 36663484032 p^{28} T^{34} + 18026833 p^{32} T^{36} - 5958 p^{36} T^{38} + p^{40} T^{40}$$
29 $$( 1 + 30 T + 151 p T^{2} + 170670 T^{3} + 10389068 T^{4} + 423361222 T^{5} + 17553493628 T^{6} + 653450347678 T^{7} + 22299617133673 T^{8} + 722949440473712 T^{9} + 21511792590651646 T^{10} + 722949440473712 p^{2} T^{11} + 22299617133673 p^{4} T^{12} + 653450347678 p^{6} T^{13} + 17553493628 p^{8} T^{14} + 423361222 p^{10} T^{15} + 10389068 p^{12} T^{16} + 170670 p^{14} T^{17} + 151 p^{17} T^{18} + 30 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
31 $$1 - 11322 T^{2} + 61135393 T^{4} - 209308330836 T^{6} + 508918901744054 T^{8} - 932077214269053926 T^{10} +$$$$13\!\cdots\!18$$$$T^{12} -$$$$15\!\cdots\!90$$$$T^{14} +$$$$14\!\cdots\!65$$$$T^{16} -$$$$12\!\cdots\!70$$$$T^{18} +$$$$11\!\cdots\!42$$$$T^{20} -$$$$12\!\cdots\!70$$$$p^{4} T^{22} +$$$$14\!\cdots\!65$$$$p^{8} T^{24} -$$$$15\!\cdots\!90$$$$p^{12} T^{26} +$$$$13\!\cdots\!18$$$$p^{16} T^{28} - 932077214269053926 p^{20} T^{30} + 508918901744054 p^{24} T^{32} - 209308330836 p^{28} T^{34} + 61135393 p^{32} T^{36} - 11322 p^{36} T^{38} + p^{40} T^{40}$$
37 $$1 - 15444 T^{2} + 121258654 T^{4} - 642109916454 T^{6} + 2565538855845221 T^{8} - 8203449137311550966 T^{10} +$$$$21\!\cdots\!21$$$$T^{12} -$$$$48\!\cdots\!42$$$$T^{14} +$$$$94\!\cdots\!54$$$$T^{16} -$$$$15\!\cdots\!58$$$$T^{18} +$$$$23\!\cdots\!14$$$$T^{20} -$$$$15\!\cdots\!58$$$$p^{4} T^{22} +$$$$94\!\cdots\!54$$$$p^{8} T^{24} -$$$$48\!\cdots\!42$$$$p^{12} T^{26} +$$$$21\!\cdots\!21$$$$p^{16} T^{28} - 8203449137311550966 p^{20} T^{30} + 2565538855845221 p^{24} T^{32} - 642109916454 p^{28} T^{34} + 121258654 p^{32} T^{36} - 15444 p^{36} T^{38} + p^{40} T^{40}$$
41 $$( 1 - 14 T + 9824 T^{2} - 242374 T^{3} + 46717743 T^{4} - 1624216260 T^{5} + 146172069565 T^{6} - 6124161123484 T^{7} + 343318573182348 T^{8} - 14982774539742700 T^{9} + 641824765638483150 T^{10} - 14982774539742700 p^{2} T^{11} + 343318573182348 p^{4} T^{12} - 6124161123484 p^{6} T^{13} + 146172069565 p^{8} T^{14} - 1624216260 p^{10} T^{15} + 46717743 p^{12} T^{16} - 242374 p^{14} T^{17} + 9824 p^{16} T^{18} - 14 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
43 $$1 - 16474 T^{2} + 141759861 T^{4} - 834293490206 T^{6} + 3736863584815476 T^{8} - 13498651110051376286 T^{10} +$$$$95\!\cdots\!05$$$$p T^{12} -$$$$10\!\cdots\!78$$$$T^{14} +$$$$24\!\cdots\!71$$$$T^{16} -$$$$51\!\cdots\!88$$$$T^{18} +$$$$99\!\cdots\!72$$$$T^{20} -$$$$51\!\cdots\!88$$$$p^{4} T^{22} +$$$$24\!\cdots\!71$$$$p^{8} T^{24} -$$$$10\!\cdots\!78$$$$p^{12} T^{26} +$$$$95\!\cdots\!05$$$$p^{17} T^{28} - 13498651110051376286 p^{20} T^{30} + 3736863584815476 p^{24} T^{32} - 834293490206 p^{28} T^{34} + 141759861 p^{32} T^{36} - 16474 p^{36} T^{38} + p^{40} T^{40}$$
47 $$1 - 17518 T^{2} + 156083133 T^{4} - 960261674600 T^{6} + 4655644696317798 T^{8} - 19025559809330919362 T^{10} +$$$$67\!\cdots\!54$$$$T^{12} -$$$$45\!\cdots\!78$$$$p T^{14} +$$$$60\!\cdots\!49$$$$T^{16} -$$$$15\!\cdots\!06$$$$T^{18} +$$$$35\!\cdots\!02$$$$T^{20} -$$$$15\!\cdots\!06$$$$p^{4} T^{22} +$$$$60\!\cdots\!49$$$$p^{8} T^{24} -$$$$45\!\cdots\!78$$$$p^{13} T^{26} +$$$$67\!\cdots\!54$$$$p^{16} T^{28} - 19025559809330919362 p^{20} T^{30} + 4655644696317798 p^{24} T^{32} - 960261674600 p^{28} T^{34} + 156083133 p^{32} T^{36} - 17518 p^{36} T^{38} + p^{40} T^{40}$$
53 $$( 1 + 4 T + 15620 T^{2} + 38666 T^{3} + 123697764 T^{4} + 2783688 T^{5} + 647196430990 T^{6} - 1415010838372 T^{7} + 2523792463046985 T^{8} - 8540842263006562 T^{9} + 7846154139849429720 T^{10} - 8540842263006562 p^{2} T^{11} + 2523792463046985 p^{4} T^{12} - 1415010838372 p^{6} T^{13} + 647196430990 p^{8} T^{14} + 2783688 p^{10} T^{15} + 123697764 p^{12} T^{16} + 38666 p^{14} T^{17} + 15620 p^{16} T^{18} + 4 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
61 $$1 - 29346 T^{2} + 453577409 T^{4} - 4851615536100 T^{6} + 40329899735365142 T^{8} -$$$$27\!\cdots\!70$$$$T^{10} +$$$$16\!\cdots\!54$$$$T^{12} -$$$$84\!\cdots\!82$$$$T^{14} +$$$$39\!\cdots\!77$$$$T^{16} -$$$$16\!\cdots\!66$$$$T^{18} +$$$$64\!\cdots\!06$$$$T^{20} -$$$$16\!\cdots\!66$$$$p^{4} T^{22} +$$$$39\!\cdots\!77$$$$p^{8} T^{24} -$$$$84\!\cdots\!82$$$$p^{12} T^{26} +$$$$16\!\cdots\!54$$$$p^{16} T^{28} -$$$$27\!\cdots\!70$$$$p^{20} T^{30} + 40329899735365142 p^{24} T^{32} - 4851615536100 p^{28} T^{34} + 453577409 p^{32} T^{36} - 29346 p^{36} T^{38} + p^{40} T^{40}$$
67 $$1 - 29100 T^{2} + 521030128 T^{4} - 6880917032148 T^{6} + 73053607440768962 T^{8} -$$$$65\!\cdots\!48$$$$T^{10} +$$$$49\!\cdots\!90$$$$T^{12} -$$$$33\!\cdots\!72$$$$T^{14} +$$$$19\!\cdots\!73$$$$T^{16} -$$$$10\!\cdots\!04$$$$T^{18} +$$$$50\!\cdots\!88$$$$T^{20} -$$$$10\!\cdots\!04$$$$p^{4} T^{22} +$$$$19\!\cdots\!73$$$$p^{8} T^{24} -$$$$33\!\cdots\!72$$$$p^{12} T^{26} +$$$$49\!\cdots\!90$$$$p^{16} T^{28} -$$$$65\!\cdots\!48$$$$p^{20} T^{30} + 73053607440768962 p^{24} T^{32} - 6880917032148 p^{28} T^{34} + 521030128 p^{32} T^{36} - 29100 p^{36} T^{38} + p^{40} T^{40}$$
71 $$( 1 - 46 T + 36049 T^{2} - 1508788 T^{3} + 619396726 T^{4} - 23137924648 T^{5} + 6756698698101 T^{6} - 223152723917514 T^{7} + 52333617071040123 T^{8} - 1520457065078166428 T^{9} +$$$$30\!\cdots\!06$$$$T^{10} - 1520457065078166428 p^{2} T^{11} + 52333617071040123 p^{4} T^{12} - 223152723917514 p^{6} T^{13} + 6756698698101 p^{8} T^{14} - 23137924648 p^{10} T^{15} + 619396726 p^{12} T^{16} - 1508788 p^{14} T^{17} + 36049 p^{16} T^{18} - 46 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
73 $$1 - 44790 T^{2} + 1067281693 T^{4} - 17526901495860 T^{6} + 221031220384497830 T^{8} -$$$$22\!\cdots\!94$$$$T^{10} +$$$$19\!\cdots\!50$$$$T^{12} -$$$$14\!\cdots\!86$$$$T^{14} +$$$$10\!\cdots\!29$$$$T^{16} -$$$$61\!\cdots\!18$$$$T^{18} +$$$$34\!\cdots\!94$$$$T^{20} -$$$$61\!\cdots\!18$$$$p^{4} T^{22} +$$$$10\!\cdots\!29$$$$p^{8} T^{24} -$$$$14\!\cdots\!86$$$$p^{12} T^{26} +$$$$19\!\cdots\!50$$$$p^{16} T^{28} -$$$$22\!\cdots\!94$$$$p^{20} T^{30} + 221031220384497830 p^{24} T^{32} - 17526901495860 p^{28} T^{34} + 1067281693 p^{32} T^{36} - 44790 p^{36} T^{38} + p^{40} T^{40}$$
79 $$( 1 + 210 T + 55013 T^{2} + 7637262 T^{3} + 1211778452 T^{4} + 130666986398 T^{5} + 15907885114307 T^{6} + 1431668961080462 T^{7} + 146389736312489335 T^{8} + 11461287473659993972 T^{9} +$$$$10\!\cdots\!80$$$$T^{10} + 11461287473659993972 p^{2} T^{11} + 146389736312489335 p^{4} T^{12} + 1431668961080462 p^{6} T^{13} + 15907885114307 p^{8} T^{14} + 130666986398 p^{10} T^{15} + 1211778452 p^{12} T^{16} + 7637262 p^{14} T^{17} + 55013 p^{16} T^{18} + 210 p^{18} T^{19} + p^{20} T^{20} )^{2}$$
83 $$1 - 77388 T^{2} + 2941221734 T^{4} - 73711040744874 T^{6} + 1376827539396685037 T^{8} -$$$$20\!\cdots\!18$$$$T^{10} +$$$$25\!\cdots\!05$$$$T^{12} -$$$$26\!\cdots\!78$$$$T^{14} +$$$$24\!\cdots\!82$$$$T^{16} -$$$$20\!\cdots\!90$$$$T^{18} +$$$$14\!\cdots\!42$$$$T^{20} -$$$$20\!\cdots\!90$$$$p^{4} T^{22} +$$$$24\!\cdots\!82$$$$p^{8} T^{24} -$$$$26\!\cdots\!78$$$$p^{12} T^{26} +$$$$25\!\cdots\!05$$$$p^{16} T^{28} -$$$$20\!\cdots\!18$$$$p^{20} T^{30} + 1376827539396685037 p^{24} T^{32} - 73711040744874 p^{28} T^{34} + 2941221734 p^{32} T^{36} - 77388 p^{36} T^{38} + p^{40} T^{40}$$
89 $$1 - 74702 T^{2} + 2706108269 T^{4} - 63436412992128 T^{6} + 1089431865086821382 T^{8} -$$$$14\!\cdots\!30$$$$T^{10} +$$$$16\!\cdots\!42$$$$T^{12} -$$$$16\!\cdots\!22$$$$T^{14} +$$$$14\!\cdots\!77$$$$T^{16} -$$$$12\!\cdots\!82$$$$T^{18} +$$$$98\!\cdots\!42$$$$T^{20} -$$$$12\!\cdots\!82$$$$p^{4} T^{22} +$$$$14\!\cdots\!77$$$$p^{8} T^{24} -$$$$16\!\cdots\!22$$$$p^{12} T^{26} +$$$$16\!\cdots\!42$$$$p^{16} T^{28} -$$$$14\!\cdots\!30$$$$p^{20} T^{30} + 1089431865086821382 p^{24} T^{32} - 63436412992128 p^{28} T^{34} + 2706108269 p^{32} T^{36} - 74702 p^{36} T^{38} + p^{40} T^{40}$$
97 $$1 - 73436 T^{2} + 3000462704 T^{4} - 86935592221236 T^{6} + 1969111993183071938 T^{8} -$$$$36\!\cdots\!80$$$$T^{10} +$$$$58\!\cdots\!98$$$$T^{12} -$$$$80\!\cdots\!88$$$$T^{14} +$$$$99\!\cdots\!05$$$$T^{16} -$$$$10\!\cdots\!28$$$$T^{18} +$$$$10\!\cdots\!08$$$$T^{20} -$$$$10\!\cdots\!28$$$$p^{4} T^{22} +$$$$99\!\cdots\!05$$$$p^{8} T^{24} -$$$$80\!\cdots\!88$$$$p^{12} T^{26} +$$$$58\!\cdots\!98$$$$p^{16} T^{28} -$$$$36\!\cdots\!80$$$$p^{20} T^{30} + 1969111993183071938 p^{24} T^{32} - 86935592221236 p^{28} T^{34} + 3000462704 p^{32} T^{36} - 73436 p^{36} T^{38} + p^{40} T^{40}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$