# Properties

 Degree 16 Conductor $2^{48} \cdot 5^{8}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 320·7-s + 376·9-s − 2.40e3·17-s + 1.76e3·23-s − 2.50e3·25-s + 3.10e4·31-s + 2.15e4·41-s + 4.76e4·47-s + 2.53e4·49-s − 1.20e5·63-s + 2.46e5·71-s + 4.64e4·73-s + 3.25e5·79-s − 2.23e4·81-s − 7.81e4·89-s + 2.49e4·97-s + 3.92e5·103-s + 2.56e5·113-s + 7.68e5·119-s + 4.61e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 9.02e5·153-s + ⋯
 L(s)  = 1 − 2.46·7-s + 1.54·9-s − 2.01·17-s + 0.693·23-s − 4/5·25-s + 5.80·31-s + 2.00·41-s + 3.14·47-s + 1.51·49-s − 3.81·63-s + 5.80·71-s + 1.01·73-s + 5.87·79-s − 0.377·81-s − 1.04·89-s + 0.269·97-s + 3.64·103-s + 1.88·113-s + 4.97·119-s + 2.86·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 3.11·153-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{48} \cdot 5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{320} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$$ $$L(3)$$ $$\approx$$ $$3.041962512$$ $$L(\frac12)$$ $$\approx$$ $$3.041962512$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$1$$
5 $$( 1 + p^{4} T^{2} )^{4}$$
good3 $$1 - 376 T^{2} + 163684 T^{4} - 4373416 p^{2} T^{6} + 146644294 p^{4} T^{8} - 4373416 p^{12} T^{10} + 163684 p^{20} T^{12} - 376 p^{30} T^{14} + p^{40} T^{16}$$
7 $$( 1 + 160 T + 25708 T^{2} + 3666960 T^{3} + 699810714 T^{4} + 3666960 p^{5} T^{5} + 25708 p^{10} T^{6} + 160 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
11 $$1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} +$$$$14\!\cdots\!94$$$$T^{8} - 9751521734727448 p^{10} T^{10} + 80897449276 p^{20} T^{12} - 461928 p^{30} T^{14} + p^{40} T^{16}$$
13 $$1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} +$$$$21\!\cdots\!70$$$$T^{8} - 514796045425358792 p^{10} T^{10} + 912998161852 p^{20} T^{12} - 1173304 p^{30} T^{14} + p^{40} T^{16}$$
17 $$( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 4006103760 p^{5} T^{5} + 4842092 p^{10} T^{6} + 1200 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
19 $$1 - 243608 p T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} +$$$$13\!\cdots\!54$$$$T^{8} - 43933554813794877112 p^{10} T^{10} + 15129915669436 p^{20} T^{12} - 243608 p^{31} T^{14} + p^{40} T^{16}$$
23 $$( 1 - 880 T + 8141852 T^{2} - 31216146720 T^{3} + 19333174339674 T^{4} - 31216146720 p^{5} T^{5} + 8141852 p^{10} T^{6} - 880 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
29 $$1 - 57762792 T^{2} + 1773990452604028 T^{4} -$$$$43\!\cdots\!44$$$$T^{6} +$$$$95\!\cdots\!70$$$$T^{8} -$$$$43\!\cdots\!44$$$$p^{10} T^{10} + 1773990452604028 p^{20} T^{12} - 57762792 p^{30} T^{14} + p^{40} T^{16}$$
31 $$( 1 - 15520 T + 133260604 T^{2} - 763025830560 T^{3} + 3983065379572806 T^{4} - 763025830560 p^{5} T^{5} + 133260604 p^{10} T^{6} - 15520 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
37 $$1 - 301131416 T^{2} + 47822074973756092 T^{4} -$$$$50\!\cdots\!08$$$$T^{6} +$$$$40\!\cdots\!70$$$$T^{8} -$$$$50\!\cdots\!08$$$$p^{10} T^{10} + 47822074973756092 p^{20} T^{12} - 301131416 p^{30} T^{14} + p^{40} T^{16}$$
41 $$( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} - 2595640517112 p^{5} T^{5} + 381689636 p^{10} T^{6} - 10792 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
43 $$1 - 542148344 T^{2} + 153567024284830372 T^{4} -$$$$32\!\cdots\!92$$$$T^{6} +$$$$53\!\cdots\!70$$$$T^{8} -$$$$32\!\cdots\!92$$$$p^{10} T^{10} + 153567024284830372 p^{20} T^{12} - 542148344 p^{30} T^{14} + p^{40} T^{16}$$
47 $$( 1 - 23840 T + 811201148 T^{2} - 13600000745040 T^{3} + 275458842496788474 T^{4} - 13600000745040 p^{5} T^{5} + 811201148 p^{10} T^{6} - 23840 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
53 $$1 - 984292984 T^{2} + 650040420165524092 T^{4} -$$$$31\!\cdots\!92$$$$T^{6} +$$$$12\!\cdots\!70$$$$T^{8} -$$$$31\!\cdots\!92$$$$p^{10} T^{10} + 650040420165524092 p^{20} T^{12} - 984292984 p^{30} T^{14} + p^{40} T^{16}$$
59 $$1 - 3042039688 T^{2} + 4949915655184301308 T^{4} -$$$$54\!\cdots\!56$$$$T^{6} +$$$$44\!\cdots\!70$$$$T^{8} -$$$$54\!\cdots\!56$$$$p^{10} T^{10} + 4949915655184301308 p^{20} T^{12} - 3042039688 p^{30} T^{14} + p^{40} T^{16}$$
61 $$1 - 2582224408 T^{2} + 4867570283316136828 T^{4} -$$$$60\!\cdots\!56$$$$T^{6} +$$$$59\!\cdots\!70$$$$T^{8} -$$$$60\!\cdots\!56$$$$p^{10} T^{10} + 4867570283316136828 p^{20} T^{12} - 2582224408 p^{30} T^{14} + p^{40} T^{16}$$
67 $$1 - 7537851704 T^{2} + 25204334234273950564 T^{4} -$$$$51\!\cdots\!16$$$$T^{6} +$$$$78\!\cdots\!94$$$$T^{8} -$$$$51\!\cdots\!16$$$$p^{10} T^{10} + 25204334234273950564 p^{20} T^{12} - 7537851704 p^{30} T^{14} + p^{40} T^{16}$$
71 $$( 1 - 123360 T + 10006413404 T^{2} - 7455069384480 p T^{3} + 24936343678058479206 T^{4} - 7455069384480 p^{6} T^{5} + 10006413404 p^{10} T^{6} - 123360 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
73 $$( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} - 115342151623520 p^{5} T^{5} + 7007019628 p^{10} T^{6} - 23200 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
79 $$( 1 - 162880 T + 14181869596 T^{2} - 858599705367360 T^{3} + 48361267115636995206 T^{4} - 858599705367360 p^{5} T^{5} + 14181869596 p^{10} T^{6} - 162880 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
83 $$1 - 29848826136 T^{2} +$$$$39\!\cdots\!24$$$$T^{4} -$$$$30\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!54$$$$T^{8} -$$$$30\!\cdots\!64$$$$p^{10} T^{10} +$$$$39\!\cdots\!24$$$$p^{20} T^{12} - 29848826136 p^{30} T^{14} + p^{40} T^{16}$$
89 $$( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} +$$$$13\!\cdots\!94$$$$T^{4} + 489497603288904 p^{5} T^{5} + 18076226204 p^{10} T^{6} + 39096 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
97 $$( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} - 824172284150080 p^{5} T^{5} + 14325999628 p^{10} T^{6} - 12480 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}