# Properties

 Degree 32 Conductor $2^{32} \cdot 3^{48}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 47·4-s + 896·13-s + 1.14e3·16-s + 2.99e4·25-s − 7.10e4·37-s + 1.09e5·49-s + 4.21e4·52-s + 7.78e4·61-s + 3.42e4·64-s − 3.80e4·73-s − 976·97-s + 1.40e6·100-s + 1.33e5·109-s − 1.48e6·121-s + 127-s + 131-s + 137-s + 139-s − 3.33e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.71e6·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 1.46·4-s + 1.47·13-s + 1.12·16-s + 9.58·25-s − 8.52·37-s + 6.53·49-s + 2.15·52-s + 2.68·61-s + 1.04·64-s − 0.835·73-s − 0.0105·97-s + 14.0·100-s + 1.07·109-s − 9.20·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 12.5·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 4.61·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{32} \cdot 3^{48}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [5/2]^{16} ),\ 1 )$$ $$L(3)$$ $$\approx$$ $$142.979$$ $$L(\frac12)$$ $$\approx$$ $$142.979$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$1 - 47 T^{2} + 265 p^{2} T^{4} - 235 p^{7} T^{6} + 745 p^{10} T^{8} - 235 p^{17} T^{10} + 265 p^{22} T^{12} - 47 p^{30} T^{14} + p^{40} T^{16}$$
3 $$1$$
good5 $$( 1 - 14972 T^{2} + 119342578 T^{4} - 617133972224 T^{6} + 90872805142579 p^{2} T^{8} - 617133972224 p^{10} T^{10} + 119342578 p^{20} T^{12} - 14972 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
7 $$( 1 - 54884 T^{2} + 1251149938 T^{4} - 15826547698400 T^{6} + 189674874534565723 T^{8} - 15826547698400 p^{10} T^{10} + 1251149938 p^{20} T^{12} - 54884 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
11 $$( 1 + 741148 T^{2} + 299964078250 T^{4} + 79656648391789648 T^{6} +$$$$15\!\cdots\!99$$$$T^{8} + 79656648391789648 p^{10} T^{10} + 299964078250 p^{20} T^{12} + 741148 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
13 $$( 1 - 224 T + 554236 T^{2} + 77263648 T^{3} + 200668790230 T^{4} + 77263648 p^{5} T^{5} + 554236 p^{10} T^{6} - 224 p^{15} T^{7} + p^{20} T^{8} )^{4}$$
17 $$( 1 - 4946936 T^{2} + 15515554849948 T^{4} - 34385249721915417800 T^{6} +$$$$55\!\cdots\!90$$$$T^{8} - 34385249721915417800 p^{10} T^{10} + 15515554849948 p^{20} T^{12} - 4946936 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
19 $$( 1 - 9713816 T^{2} + 55349997016060 T^{4} -$$$$21\!\cdots\!72$$$$T^{6} +$$$$61\!\cdots\!02$$$$T^{8} -$$$$21\!\cdots\!72$$$$p^{10} T^{10} + 55349997016060 p^{20} T^{12} - 9713816 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
23 $$( 1 + 29994040 T^{2} + 487786705761244 T^{4} +$$$$51\!\cdots\!44$$$$T^{6} +$$$$39\!\cdots\!14$$$$T^{8} +$$$$51\!\cdots\!44$$$$p^{10} T^{10} + 487786705761244 p^{20} T^{12} + 29994040 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
29 $$( 1 - 26575640 T^{2} + 859790443267708 T^{4} -$$$$25\!\cdots\!88$$$$T^{6} +$$$$44\!\cdots\!26$$$$T^{8} -$$$$25\!\cdots\!88$$$$p^{10} T^{10} + 859790443267708 p^{20} T^{12} - 26575640 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
31 $$( 1 - 160216820 T^{2} + 12412163878108546 T^{4} -$$$$60\!\cdots\!08$$$$T^{6} +$$$$20\!\cdots\!87$$$$T^{8} -$$$$60\!\cdots\!08$$$$p^{10} T^{10} + 12412163878108546 p^{20} T^{12} - 160216820 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
37 $$( 1 + 17752 T + 254119924 T^{2} + 2137611581224 T^{3} + 19827261455867542 T^{4} + 2137611581224 p^{5} T^{5} + 254119924 p^{10} T^{6} + 17752 p^{15} T^{7} + p^{20} T^{8} )^{4}$$
41 $$( 1 - 667999304 T^{2} + 215857510496501980 T^{4} -$$$$43\!\cdots\!84$$$$T^{6} +$$$$60\!\cdots\!14$$$$T^{8} -$$$$43\!\cdots\!84$$$$p^{10} T^{10} + 215857510496501980 p^{20} T^{12} - 667999304 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
43 $$( 1 - 271892936 T^{2} + 81822516500494204 T^{4} -$$$$14\!\cdots\!72$$$$T^{6} +$$$$26\!\cdots\!02$$$$T^{8} -$$$$14\!\cdots\!72$$$$p^{10} T^{10} + 81822516500494204 p^{20} T^{12} - 271892936 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
47 $$( 1 + 1268760328 T^{2} + 775645167851419804 T^{4} +$$$$30\!\cdots\!92$$$$T^{6} +$$$$81\!\cdots\!58$$$$T^{8} +$$$$30\!\cdots\!92$$$$p^{10} T^{10} + 775645167851419804 p^{20} T^{12} + 1268760328 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
53 $$( 1 - 1665548972 T^{2} + 1494189343182800962 T^{4} -$$$$95\!\cdots\!20$$$$T^{6} +$$$$46\!\cdots\!75$$$$T^{8} -$$$$95\!\cdots\!20$$$$p^{10} T^{10} + 1494189343182800962 p^{20} T^{12} - 1665548972 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
59 $$( 1 + 4822767208 T^{2} + 10641078380871180220 T^{4} +$$$$14\!\cdots\!92$$$$T^{6} +$$$$12\!\cdots\!22$$$$T^{8} +$$$$14\!\cdots\!92$$$$p^{10} T^{10} + 10641078380871180220 p^{20} T^{12} + 4822767208 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
61 $$( 1 - 19472 T + 1261489684 T^{2} - 43949943879536 T^{3} + 833752337244254902 T^{4} - 43949943879536 p^{5} T^{5} + 1261489684 p^{10} T^{6} - 19472 p^{15} T^{7} + p^{20} T^{8} )^{4}$$
67 $$( 1 - 1447449416 T^{2} + 2915553445416739516 T^{4} -$$$$52\!\cdots\!68$$$$T^{6} +$$$$74\!\cdots\!58$$$$T^{8} -$$$$52\!\cdots\!68$$$$p^{10} T^{10} + 2915553445416739516 p^{20} T^{12} - 1447449416 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
71 $$( 1 + 9830208232 T^{2} + 46886702724166517020 T^{4} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$30\!\cdots\!22$$$$T^{8} +$$$$14\!\cdots\!88$$$$p^{10} T^{10} + 46886702724166517020 p^{20} T^{12} + 9830208232 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
73 $$( 1 + 9508 T + 3097043794 T^{2} - 21796113866240 T^{3} + 7953703459188333211 T^{4} - 21796113866240 p^{5} T^{5} + 3097043794 p^{10} T^{6} + 9508 p^{15} T^{7} + p^{20} T^{8} )^{4}$$
79 $$( 1 + 99394360 T^{2} + 19778301042529431580 T^{4} -$$$$22\!\cdots\!96$$$$T^{6} +$$$$20\!\cdots\!66$$$$T^{8} -$$$$22\!\cdots\!96$$$$p^{10} T^{10} + 19778301042529431580 p^{20} T^{12} + 99394360 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
83 $$( 1 + 20559730876 T^{2} +$$$$21\!\cdots\!14$$$$T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$70\!\cdots\!39$$$$T^{8} +$$$$14\!\cdots\!68$$$$p^{10} T^{10} +$$$$21\!\cdots\!14$$$$p^{20} T^{12} + 20559730876 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
89 $$( 1 - 30119375768 T^{2} +$$$$43\!\cdots\!12$$$$T^{4} -$$$$41\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!18$$$$T^{8} -$$$$41\!\cdots\!60$$$$p^{10} T^{10} +$$$$43\!\cdots\!12$$$$p^{20} T^{12} - 30119375768 p^{30} T^{14} + p^{40} T^{16} )^{2}$$
97 $$( 1 + 244 T + 6580710202 T^{2} + 1109461517844208 T^{3} - 18256445891874654653 T^{4} + 1109461517844208 p^{5} T^{5} + 6580710202 p^{10} T^{6} + 244 p^{15} T^{7} + p^{20} T^{8} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}