# Properties

 Degree 20 Conductor $2^{100}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 28·7-s − 108·9-s + 4·17-s + 276·23-s − 600·25-s + 368·31-s + 944·47-s − 1.37e3·49-s − 3.02e3·63-s + 3.46e3·71-s − 296·73-s + 4.41e3·79-s + 5.34e3·81-s + 88·89-s − 4·97-s + 6.95e3·103-s + 668·113-s + 112·119-s − 7.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 432·153-s + 157-s + ⋯
 L(s)  = 1 + 1.51·7-s − 4·9-s + 0.0570·17-s + 2.50·23-s − 4.79·25-s + 2.13·31-s + 2.92·47-s − 3.99·49-s − 6.04·63-s + 5.79·71-s − 0.474·73-s + 6.28·79-s + 7.33·81-s + 0.104·89-s − 0.00418·97-s + 6.65·103-s + 0.556·113-s + 0.0862·119-s − 5.45·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.228·153-s + 0.000508·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$20$$ $$N$$ = $$2^{100}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{1024} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(20,\ 2^{100} ,\ ( \ : [3/2]^{10} ),\ 1 )$$ $$L(2)$$ $$\approx$$ $$0.4506368658$$ $$L(\frac12)$$ $$\approx$$ $$0.4506368658$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 20. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 4 p^{3} T^{2} + 6317 T^{4} + 275312 T^{6} + 3302590 p T^{8} + 295366792 T^{10} + 3302590 p^{7} T^{12} + 275312 p^{12} T^{14} + 6317 p^{18} T^{16} + 4 p^{27} T^{18} + p^{30} T^{20}$$
5 $$1 + 24 p^{2} T^{2} + 191789 T^{4} + 42513504 T^{6} + 7224666922 T^{8} + 994659832592 T^{10} + 7224666922 p^{6} T^{12} + 42513504 p^{12} T^{14} + 191789 p^{18} T^{16} + 24 p^{26} T^{18} + p^{30} T^{20}$$
7 $$( 1 - 2 p T + 979 T^{2} - 5832 T^{3} + 372410 T^{4} - 662580 T^{5} + 372410 p^{3} T^{6} - 5832 p^{6} T^{7} + 979 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} )^{2}$$
11 $$1 + 60 p^{2} T^{2} + 24476093 T^{4} + 51189622576 T^{6} + 78543614693626 T^{8} + 105919424238644520 T^{10} + 78543614693626 p^{6} T^{12} + 51189622576 p^{12} T^{14} + 24476093 p^{18} T^{16} + 60 p^{26} T^{18} + p^{30} T^{20}$$
13 $$1 + 72 p^{2} T^{2} + 76341149 T^{4} + 324341675296 T^{6} + 1028186938895082 T^{8} + 2540163312339385904 T^{10} + 1028186938895082 p^{6} T^{12} + 324341675296 p^{12} T^{14} + 76341149 p^{18} T^{16} + 72 p^{26} T^{18} + 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 + 33948 T^{2} + 631492045 T^{4} + 8119820572464 T^{6} + 79259035180229178 T^{8} +$$$$60\!\cdots\!76$$$$T^{10} + 79259035180229178 p^{6} T^{12} + 8119820572464 p^{12} T^{14} + 631492045 p^{18} T^{16} + 33948 p^{24} T^{18} + p^{30} T^{20}$$
23 $$( 1 - 6 p T + 47715 T^{2} - 4085400 T^{3} + 882303386 T^{4} - 56970028764 T^{5} + 882303386 p^{3} T^{6} - 4085400 p^{6} T^{7} + 47715 p^{9} T^{8} - 6 p^{13} T^{9} + p^{15} T^{10} )^{2}$$
29 $$1 + 154168 T^{2} + 380663905 p T^{4} + 493373494152672 T^{6} + 16008852482124057194 T^{8} +$$$$42\!\cdots\!28$$$$T^{10} + 16008852482124057194 p^{6} T^{12} + 493373494152672 p^{12} T^{14} + 380663905 p^{19} T^{16} + 154168 p^{24} T^{18} + p^{30} T^{20}$$
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 $$1 + 361064 T^{2} + 63572869101 T^{4} + 7160954906480288 T^{6} +$$$$56\!\cdots\!06$$$$T^{8} +$$$$33\!\cdots\!92$$$$T^{10} +$$$$56\!\cdots\!06$$$$p^{6} T^{12} + 7160954906480288 p^{12} T^{14} + 63572869101 p^{18} T^{16} + 361064 p^{24} T^{18} + p^{30} T^{20}$$
41 $$( 1 + 220509 T^{2} + 10715136 T^{3} + 24270968490 T^{4} + 1235215122432 T^{5} + 24270968490 p^{3} T^{6} + 10715136 p^{6} T^{7} + 220509 p^{9} T^{8} + p^{15} T^{10} )^{2}$$
43 $$1 + 12836 p T^{2} + 145009588605 T^{4} + 24152295654927088 T^{6} +$$$$28\!\cdots\!30$$$$T^{8} +$$$$25\!\cdots\!12$$$$T^{10} +$$$$28\!\cdots\!30$$$$p^{6} T^{12} + 24152295654927088 p^{12} T^{14} + 145009588605 p^{18} T^{16} + 12836 p^{25} T^{18} + p^{30} T^{20}$$
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 $$1 + 525160 T^{2} + 112387404877 T^{4} + 12025532417207968 T^{6} +$$$$67\!\cdots\!30$$$$T^{8} +$$$$40\!\cdots\!00$$$$T^{10} +$$$$67\!\cdots\!30$$$$p^{6} T^{12} + 12025532417207968 p^{12} T^{14} + 112387404877 p^{18} T^{16} + 525160 p^{24} T^{18} + p^{30} T^{20}$$
59 $$1 + 958284 T^{2} + 532967592733 T^{4} + 207330855222256112 T^{6} +$$$$61\!\cdots\!62$$$$T^{8} +$$$$40\!\cdots\!88$$$$p^{2} T^{10} +$$$$61\!\cdots\!62$$$$p^{6} T^{12} + 207330855222256112 p^{12} T^{14} + 532967592733 p^{18} T^{16} + 958284 p^{24} T^{18} + p^{30} T^{20}$$
61 $$1 + 1146824 T^{2} + 8354214273 p T^{4} + 80080659050426912 T^{6} -$$$$14\!\cdots\!54$$$$T^{8} -$$$$80\!\cdots\!72$$$$T^{10} -$$$$14\!\cdots\!54$$$$p^{6} T^{12} + 80080659050426912 p^{12} T^{14} + 8354214273 p^{19} T^{16} + 1146824 p^{24} T^{18} + p^{30} T^{20}$$
67 $$1 + 2130652 T^{2} + 2127009110445 T^{4} + 1334413095892987440 T^{6} +$$$$59\!\cdots\!58$$$$T^{8} +$$$$20\!\cdots\!52$$$$T^{10} +$$$$59\!\cdots\!58$$$$p^{6} T^{12} + 1334413095892987440 p^{12} T^{14} + 2127009110445 p^{18} T^{16} + 2130652 p^{24} T^{18} + p^{30} T^{20}$$
71 $$( 1 - 1734 T + 2753587 T^{2} - 2611066280 T^{3} + 2272726706522 T^{4} - 1414434499369348 T^{5} + 2272726706522 p^{3} T^{6} - 2611066280 p^{6} T^{7} + 2753587 p^{9} T^{8} - 1734 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
73 $$( 1 + 148 T + 1578093 T^{2} + 253463696 T^{3} + 1105903969594 T^{4} + 150341055768952 T^{5} + 1105903969594 p^{3} T^{6} + 253463696 p^{6} T^{7} + 1578093 p^{9} T^{8} + 148 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
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 $$1 + 2541516 T^{2} + 3317268679373 T^{4} + 33811498843796816 p T^{6} +$$$$18\!\cdots\!66$$$$T^{8} +$$$$10\!\cdots\!48$$$$T^{10} +$$$$18\!\cdots\!66$$$$p^{6} T^{12} + 33811498843796816 p^{13} T^{14} + 3317268679373 p^{18} T^{16} + 2541516 p^{24} T^{18} + p^{30} T^{20}$$
89 $$( 1 - 44 T + 1822557 T^{2} - 625587056 T^{3} + 1619388326906 T^{4} - 839474353817096 T^{5} + 1619388326906 p^{3} T^{6} - 625587056 p^{6} T^{7} + 1822557 p^{9} T^{8} - 44 p^{12} T^{9} + p^{15} T^{10} )^{2}$$
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}