# Properties

 Degree 20 Conductor $2^{40}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 2·3-s + 6·4-s − 2·5-s + 4·6-s − 24·8-s + 2·9-s + 4·10-s + 18·11-s − 12·12-s − 2·13-s + 4·15-s + 116·16-s − 4·17-s − 4·18-s − 26·19-s − 12·20-s − 36·22-s + 48·24-s + 2·25-s + 4·26-s + 42·27-s − 202·29-s − 8·30-s + 368·31-s − 56·32-s − 36·33-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.384·3-s + 3/4·4-s − 0.178·5-s + 0.272·6-s − 1.06·8-s + 2/27·9-s + 0.126·10-s + 0.493·11-s − 0.288·12-s − 0.0426·13-s + 0.0688·15-s + 1.81·16-s − 0.0570·17-s − 0.0523·18-s − 0.313·19-s − 0.134·20-s − 0.348·22-s + 0.408·24-s + 0.0159·25-s + 0.0301·26-s + 0.299·27-s − 1.29·29-s − 0.0486·30-s + 2.13·31-s − 0.309·32-s − 0.189·33-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$20$$ $$N$$ = $$2^{40}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{16} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(20,\ 2^{40} ,\ ( \ : [3/2]^{10} ),\ 1 )$ $L(2)$ $\approx$ $0.637296$ $L(\frac12)$ $\approx$ $0.637296$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 20. If $p = 2$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 $$1 + p T - p T^{2} + p^{3} T^{3} - 5 p^{3} T^{4} - 11 p^{5} T^{5} - 5 p^{6} T^{6} + p^{9} T^{7} - p^{10} T^{8} + p^{13} T^{9} + p^{15} T^{10}$$
good3 $$1 + 2 T + 2 T^{2} - 14 p T^{3} - 571 T^{4} + 760 T^{5} + 3544 T^{6} + 22264 p T^{7} + 57874 T^{8} - 3046228 T^{9} - 6701044 T^{10} - 3046228 p^{3} T^{11} + 57874 p^{6} T^{12} + 22264 p^{10} T^{13} + 3544 p^{12} T^{14} + 760 p^{15} T^{15} - 571 p^{18} T^{16} - 14 p^{22} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20}$$
5 $$1 + 2 T + 2 T^{2} - 966 T^{3} - 13723 T^{4} + 3608 p T^{5} + 530104 T^{6} - 11981288 T^{7} - 28535006 T^{8} + 2301854348 T^{9} + 23672040908 T^{10} + 2301854348 p^{3} T^{11} - 28535006 p^{6} T^{12} - 11981288 p^{9} T^{13} + 530104 p^{12} T^{14} + 3608 p^{16} T^{15} - 13723 p^{18} T^{16} - 966 p^{21} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20}$$
7 $$1 - 1762 T^{2} + 219995 p T^{4} - 133155368 p T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 440869947922 p^{6} T^{12} - 133155368 p^{13} T^{14} + 219995 p^{19} T^{16} - 1762 p^{24} T^{18} + p^{30} T^{20}$$
11 $$1 - 18 T + 162 T^{2} - 122934 T^{3} + 4077397 T^{4} + 79597000 T^{5} + 5463099864 T^{6} - 313798751208 T^{7} - 56924680478 p^{2} T^{8} + 9237744161500 p T^{9} + 18755914132083020 T^{10} + 9237744161500 p^{4} T^{11} - 56924680478 p^{8} T^{12} - 313798751208 p^{9} T^{13} + 5463099864 p^{12} T^{14} + 79597000 p^{15} T^{15} + 4077397 p^{18} T^{16} - 122934 p^{21} T^{17} + 162 p^{24} T^{18} - 18 p^{27} T^{19} + p^{30} T^{20}$$
13 $$1 + 2 T + 2 T^{2} - 45206 T^{3} - 184727 p T^{4} + 29395960 T^{5} + 1085386040 T^{6} - 29556007880 p T^{7} - 9400034983966 T^{8} + 1531455561616908 T^{9} + 23489689415409228 T^{10} + 1531455561616908 p^{3} T^{11} - 9400034983966 p^{6} T^{12} - 29556007880 p^{10} T^{13} + 1085386040 p^{12} T^{14} + 29395960 p^{15} T^{15} - 184727 p^{19} T^{16} - 45206 p^{21} T^{17} + 2 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20}$$
17 $$( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 98460610 p^{3} T^{6} + 102520 p^{6} T^{7} + 12653 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
19 $$1 + 26 T + 338 T^{2} - 339906 T^{3} - 64153371 T^{4} + 2461461784 T^{5} + 143449890200 T^{6} + 36783398837960 T^{7} + 1011857007777554 T^{8} - 259766590630759364 T^{9} - 7366645907488092948 T^{10} - 259766590630759364 p^{3} T^{11} + 1011857007777554 p^{6} T^{12} + 36783398837960 p^{9} T^{13} + 143449890200 p^{12} T^{14} + 2461461784 p^{15} T^{15} - 64153371 p^{18} T^{16} - 339906 p^{21} T^{17} + 338 p^{24} T^{18} + 26 p^{27} T^{19} + p^{30} T^{20}$$
23 $$1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} -$$$$18\!\cdots\!84$$$$T^{10} + 1342371312768300946 p^{6} T^{12} - 73253961622040 p^{12} T^{14} + 2913757597 p^{18} T^{16} - 76386 p^{24} T^{18} + p^{30} T^{20}$$
29 $$1 + 202 T + 20402 T^{2} - 1177934 T^{3} + 398569397 T^{4} + 239164019416 T^{5} + 40873283338616 T^{6} + 2529271278095288 T^{7} + 194871598558001506 T^{8} +$$$$12\!\cdots\!32$$$$T^{9} +$$$$30\!\cdots\!36$$$$T^{10} +$$$$12\!\cdots\!32$$$$p^{3} T^{11} + 194871598558001506 p^{6} T^{12} + 2529271278095288 p^{9} T^{13} + 40873283338616 p^{12} T^{14} + 239164019416 p^{15} T^{15} + 398569397 p^{18} T^{16} - 1177934 p^{21} T^{17} + 20402 p^{24} T^{18} + 202 p^{27} T^{19} + p^{30} T^{20}$$
31 $$( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 7638677322 p^{3} T^{6} - 19809056 p^{6} T^{7} + 134043 p^{9} T^{8} - 184 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
37 $$1 + 10 T + 50 T^{2} + 1972962 T^{3} + 1630465317 T^{4} - 153991562664 T^{5} + 324850634232 T^{6} - 5252842710654600 T^{7} + 3474549392106364962 T^{8} + 27985624577691139772 T^{9} +$$$$10\!\cdots\!24$$$$T^{10} + 27985624577691139772 p^{3} T^{11} + 3474549392106364962 p^{6} T^{12} - 5252842710654600 p^{9} T^{13} + 324850634232 p^{12} T^{14} - 153991562664 p^{15} T^{15} + 1630465317 p^{18} T^{16} + 1972962 p^{21} T^{17} + 50 p^{24} T^{18} + 10 p^{27} T^{19} + p^{30} T^{20}$$
41 $$1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} +$$$$14\!\cdots\!14$$$$T^{8} -$$$$11\!\cdots\!88$$$$T^{10} +$$$$14\!\cdots\!14$$$$p^{6} T^{12} - 13934678680622904 p^{12} T^{14} + 97166156061 p^{18} T^{16} - 441018 p^{24} T^{18} + p^{30} T^{20}$$
43 $$1 + 838 T + 351122 T^{2} + 132133650 T^{3} + 56398378005 T^{4} + 20936462157416 T^{5} + 6471694737204248 T^{6} + 1952595983380873720 T^{7} +$$$$59\!\cdots\!10$$$$T^{8} +$$$$17\!\cdots\!68$$$$T^{9} +$$$$49\!\cdots\!52$$$$T^{10} +$$$$17\!\cdots\!68$$$$p^{3} T^{11} +$$$$59\!\cdots\!10$$$$p^{6} T^{12} + 1952595983380873720 p^{9} T^{13} + 6471694737204248 p^{12} T^{14} + 20936462157416 p^{15} T^{15} + 56398378005 p^{18} T^{16} + 132133650 p^{21} T^{17} + 351122 p^{24} T^{18} + 838 p^{27} T^{19} + p^{30} T^{20}$$
47 $$( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 90105579914 p^{3} T^{6} + 171516064 p^{6} T^{7} + 462219 p^{9} T^{8} + 472 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
53 $$1 + 378 T + 71442 T^{2} - 995350 p T^{3} + 3016286341 T^{4} - 526686651752 T^{5} + 976891435665272 T^{6} - 1674349213754452168 T^{7} - 1581505854923305054 T^{8} -$$$$14\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!08$$$$T^{10} -$$$$14\!\cdots\!20$$$$p^{3} T^{11} - 1581505854923305054 p^{6} T^{12} - 1674349213754452168 p^{9} T^{13} + 976891435665272 p^{12} T^{14} - 526686651752 p^{15} T^{15} + 3016286341 p^{18} T^{16} - 995350 p^{22} T^{17} + 71442 p^{24} T^{18} + 378 p^{27} T^{19} + p^{30} T^{20}$$
59 $$1 - 1706 T + 1455218 T^{2} - 989315358 T^{3} + 555128806581 T^{4} - 232658117164632 T^{5} + 78453755015006616 T^{6} - 20092577710244830152 T^{7} +$$$$57\!\cdots\!98$$$$T^{8} +$$$$37\!\cdots\!12$$$$p T^{9} -$$$$35\!\cdots\!80$$$$p^{2} T^{10} +$$$$37\!\cdots\!12$$$$p^{4} T^{11} +$$$$57\!\cdots\!98$$$$p^{6} T^{12} - 20092577710244830152 p^{9} T^{13} + 78453755015006616 p^{12} T^{14} - 232658117164632 p^{15} T^{15} + 555128806581 p^{18} T^{16} - 989315358 p^{21} T^{17} + 1455218 p^{24} T^{18} - 1706 p^{27} T^{19} + p^{30} T^{20}$$
61 $$1 - 910 T + 414050 T^{2} + 45940410 T^{3} + 20471098485 T^{4} - 72823214590920 T^{5} + 58848327585507000 T^{6} - 6001380717052735080 T^{7} +$$$$59\!\cdots\!50$$$$T^{8} -$$$$32\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!00$$$$T^{10} -$$$$32\!\cdots\!00$$$$p^{3} T^{11} +$$$$59\!\cdots\!50$$$$p^{6} T^{12} - 6001380717052735080 p^{9} T^{13} + 58848327585507000 p^{12} T^{14} - 72823214590920 p^{15} T^{15} + 20471098485 p^{18} T^{16} + 45940410 p^{21} T^{17} + 414050 p^{24} T^{18} - 910 p^{27} T^{19} + p^{30} T^{20}$$
67 $$1 - 1942 T + 1885682 T^{2} - 1530965298 T^{3} + 1338066168261 T^{4} - 1068751594464168 T^{5} + 724275679990789656 T^{6} -$$$$46\!\cdots\!12$$$$T^{7} +$$$$29\!\cdots\!06$$$$T^{8} -$$$$17\!\cdots\!64$$$$T^{9} +$$$$97\!\cdots\!68$$$$T^{10} -$$$$17\!\cdots\!64$$$$p^{3} T^{11} +$$$$29\!\cdots\!06$$$$p^{6} T^{12} -$$$$46\!\cdots\!12$$$$p^{9} T^{13} + 724275679990789656 p^{12} T^{14} - 1068751594464168 p^{15} T^{15} + 1338066168261 p^{18} T^{16} - 1530965298 p^{21} T^{17} + 1885682 p^{24} T^{18} - 1942 p^{27} T^{19} + p^{30} T^{20}$$
71 $$1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} +$$$$13\!\cdots\!66$$$$T^{8} -$$$$55\!\cdots\!32$$$$T^{10} +$$$$13\!\cdots\!66$$$$p^{6} T^{12} - 2420243241413642648 p^{12} T^{14} + 3072516920573 p^{18} T^{16} - 2500418 p^{24} T^{18} + p^{30} T^{20}$$
73 $$1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} +$$$$26\!\cdots\!74$$$$T^{8} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$26\!\cdots\!74$$$$p^{6} T^{12} - 4242124717558516472 p^{12} T^{14} + 4627160201821 p^{18} T^{16} - 3134282 p^{24} T^{18} + p^{30} T^{20}$$
79 $$( 1 + 2208 T + 3816107 T^{2} + 54694528 p T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 4245684014154 p^{3} T^{6} + 54694528 p^{7} T^{7} + 3816107 p^{9} T^{8} + 2208 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
83 $$1 + 2562 T + 3281922 T^{2} + 2891460918 T^{3} + 1934799974629 T^{4} + 1191348439341176 T^{5} + 882645219418437336 T^{6} +$$$$79\!\cdots\!28$$$$T^{7} +$$$$89\!\cdots\!74$$$$T^{8} +$$$$97\!\cdots\!88$$$$T^{9} +$$$$82\!\cdots\!76$$$$T^{10} +$$$$97\!\cdots\!88$$$$p^{3} T^{11} +$$$$89\!\cdots\!74$$$$p^{6} T^{12} +$$$$79\!\cdots\!28$$$$p^{9} T^{13} + 882645219418437336 p^{12} T^{14} + 1191348439341176 p^{15} T^{15} + 1934799974629 p^{18} T^{16} + 2891460918 p^{21} T^{17} + 3281922 p^{24} T^{18} + 2562 p^{27} T^{19} + p^{30} T^{20}$$
89 $$1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} +$$$$69\!\cdots\!98$$$$T^{8} -$$$$52\!\cdots\!28$$$$T^{10} +$$$$69\!\cdots\!98$$$$p^{6} T^{12} - 7720859292932177528 p^{12} T^{14} + 6505439011133 p^{18} T^{16} - 3643178 p^{24} T^{18} + p^{30} T^{20}$$
97 $$( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}