# Properties

 Degree 32 Conductor $2^{32} \cdot 3^{48}$ Sign $1$ Motivic weight 4 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 7·4-s − 352·13-s − 27·16-s − 4.18e3·25-s + 3.20e3·37-s + 1.87e4·49-s + 2.46e3·52-s − 2.75e3·61-s + 417·64-s + 8.24e3·73-s − 6.92e3·97-s + 2.92e4·100-s − 1.69e4·109-s + 9.60e4·121-s + 127-s + 131-s + 137-s + 139-s − 2.24e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.76e5·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 − 0.437·4-s − 2.08·13-s − 0.105·16-s − 6.69·25-s + 2.33·37-s + 7.81·49-s + 0.911·52-s − 0.739·61-s + 0.101·64-s + 1.54·73-s − 0.736·97-s + 2.92·100-s − 1.42·109-s + 6.56·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 1.02·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 6.18·169-s + 3.34e−5·173-s + 3.12e−5·179-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(5-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{32} \cdot 3^{48}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{32} \cdot 3^{48} ,\ ( \ : [2]^{16} ),\ 1 )$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.29603$$ $$L(\frac12)$$ $$\approx$$ $$1.29603$$ $$L(3)$$ 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 + 7 T^{2} + 19 p^{2} T^{4} + 19 p^{4} T^{6} + 185 p^{9} T^{8} + 19 p^{12} T^{10} + 19 p^{18} T^{12} + 7 p^{24} T^{14} + p^{32} T^{16}$$
3 $$1$$
good5 $$( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 1622274208 p^{8} T^{10} + 2149522 p^{16} T^{12} + 2092 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
7 $$( 1 - 1340 p T^{2} + 36105250 T^{4} - 10452636224 p T^{6} + 129372615567787 T^{8} - 10452636224 p^{9} T^{10} + 36105250 p^{16} T^{12} - 1340 p^{25} T^{14} + p^{32} T^{16} )^{2}$$
11 $$( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} - 20151290866448 p^{8} T^{10} + 1167571642 p^{16} T^{12} - 48044 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
13 $$( 1 + 88 T + 63508 T^{2} + 390568 p T^{3} + 2650180198 T^{4} + 390568 p^{5} T^{5} + 63508 p^{8} T^{6} + 88 p^{12} T^{7} + p^{16} T^{8} )^{4}$$
17 $$( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} +$$$$47\!\cdots\!02$$$$T^{8} + 5059782297489112 p^{8} T^{10} + 41438060380 p^{16} T^{12} + 239464 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
19 $$( 1 - 680120 T^{2} + 646972732 p^{2} T^{4} - 51394020863660552 T^{6} +$$$$79\!\cdots\!98$$$$T^{8} - 51394020863660552 p^{8} T^{10} + 646972732 p^{18} T^{12} - 680120 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
23 $$( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} +$$$$35\!\cdots\!18$$$$T^{8} - 115630955446337192 p^{8} T^{10} + 314533671388 p^{16} T^{12} - 675608 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
29 $$( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} +$$$$44\!\cdots\!14$$$$T^{8} + 5416371986440228072 p^{8} T^{10} + 4863922457692 p^{16} T^{12} + 2998744 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
31 $$( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} +$$$$29\!\cdots\!59$$$$T^{8} - 2424947225563311392 p^{8} T^{10} + 2683287954034 p^{16} T^{12} - 1328756 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
37 $$( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 3784975328 p^{4} T^{5} + 1852732 p^{8} T^{6} - 800 p^{12} T^{7} + p^{16} T^{8} )^{4}$$
41 $$( 1 + 14045080 T^{2} + 100482710328412 T^{4} +$$$$46\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!54$$$$T^{8} +$$$$46\!\cdots\!84$$$$p^{8} T^{10} + 100482710328412 p^{16} T^{12} + 14045080 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
43 $$( 1 - 11990648 T^{2} + 69115070774812 T^{4} -$$$$29\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!18$$$$T^{8} -$$$$29\!\cdots\!24$$$$p^{8} T^{10} + 69115070774812 p^{16} T^{12} - 11990648 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
47 $$( 1 - 27701336 T^{2} + 374201000701276 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{8} -$$$$31\!\cdots\!48$$$$p^{8} T^{10} + 374201000701276 p^{16} T^{12} - 27701336 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
53 $$( 1 + 35837884 T^{2} + 582188420839618 T^{4} +$$$$60\!\cdots\!16$$$$T^{6} +$$$$50\!\cdots\!79$$$$T^{8} +$$$$60\!\cdots\!16$$$$p^{8} T^{10} + 582188420839618 p^{16} T^{12} + 35837884 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
59 $$( 1 - 70312136 T^{2} + 2419955263154716 T^{4} -$$$$51\!\cdots\!68$$$$T^{6} +$$$$75\!\cdots\!74$$$$T^{8} -$$$$51\!\cdots\!68$$$$p^{8} T^{10} + 2419955263154716 p^{16} T^{12} - 70312136 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
61 $$( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 28994874640 p^{4} T^{5} + 35342980 p^{8} T^{6} + 688 p^{12} T^{7} + p^{16} T^{8} )^{4}$$
67 $$( 1 - 29579000 T^{2} + 1274223737389084 T^{4} -$$$$30\!\cdots\!28$$$$T^{6} +$$$$73\!\cdots\!58$$$$T^{8} -$$$$30\!\cdots\!28$$$$p^{8} T^{10} + 1274223737389084 p^{16} T^{12} - 29579000 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
71 $$( 1 - 86649128 T^{2} + 3567527319446236 T^{4} -$$$$11\!\cdots\!36$$$$T^{6} +$$$$32\!\cdots\!90$$$$T^{8} -$$$$11\!\cdots\!36$$$$p^{8} T^{10} + 3567527319446236 p^{16} T^{12} - 86649128 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
73 $$( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 101224244576 p^{4} T^{5} + 55108594 p^{8} T^{6} - 2060 p^{12} T^{7} + p^{16} T^{8} )^{4}$$
79 $$( 1 - 162885800 T^{2} + 12409200314583772 T^{4} -$$$$63\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!06$$$$T^{8} -$$$$63\!\cdots\!36$$$$p^{8} T^{10} + 12409200314583772 p^{16} T^{12} - 162885800 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
83 $$( 1 - 140155436 T^{2} + 8287556309932090 T^{4} -$$$$42\!\cdots\!92$$$$T^{6} +$$$$22\!\cdots\!03$$$$T^{8} -$$$$42\!\cdots\!92$$$$p^{8} T^{10} + 8287556309932090 p^{16} T^{12} - 140155436 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
89 $$( 1 + 257253400 T^{2} + 38619499284608860 T^{4} +$$$$38\!\cdots\!44$$$$T^{6} +$$$$28\!\cdots\!58$$$$T^{8} +$$$$38\!\cdots\!44$$$$p^{8} T^{10} + 38619499284608860 p^{16} T^{12} + 257253400 p^{24} T^{14} + p^{32} T^{16} )^{2}$$
97 $$( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 428249450896 p^{4} T^{5} + 297359818 p^{8} T^{6} + 1732 p^{12} T^{7} + p^{16} T^{8} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}