# Properties

 Degree 32 Conductor $2^{80} \cdot 7^{16}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·9-s − 48·17-s − 60·25-s − 16·29-s + 72·37-s + 64·49-s + 40·53-s + 216·61-s − 312·73-s + 122·81-s + 24·89-s − 480·101-s + 112·109-s + 1.32e3·113-s + 432·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 768·153-s + 157-s + 163-s + 167-s + 2.11e3·169-s + 173-s + ⋯
 L(s)  = 1 − 1.77·9-s − 2.82·17-s − 2.39·25-s − 0.551·29-s + 1.94·37-s + 1.30·49-s + 0.754·53-s + 3.54·61-s − 4.27·73-s + 1.50·81-s + 0.269·89-s − 4.75·101-s + 1.02·109-s + 11.7·113-s + 3.57·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 5.01·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 12.4·169-s + 0.00578·173-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{80} \cdot 7^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{224} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{80} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$6.72481$$ $$L(\frac12)$$ $$\approx$$ $$6.72481$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$1$$
7 $$1 - 64 T^{2} + 324 p T^{4} + 64 p^{2} T^{6} - 2938 p^{4} T^{8} + 64 p^{6} T^{10} + 324 p^{9} T^{12} - 64 p^{12} T^{14} + p^{16} T^{16}$$
good3 $$1 + 16 T^{2} + 134 T^{4} + 1120 T^{6} + 7897 T^{8} + 73856 T^{10} + 1194422 T^{12} + 13226000 T^{14} + 117489172 T^{16} + 13226000 p^{4} T^{18} + 1194422 p^{8} T^{20} + 73856 p^{12} T^{22} + 7897 p^{16} T^{24} + 1120 p^{20} T^{26} + 134 p^{24} T^{28} + 16 p^{28} T^{30} + p^{32} T^{32}$$
5 $$( 1 + 6 p T^{2} + 321 T^{4} + 48 p^{2} T^{5} - 186 p T^{6} + 27792 p T^{7} - 137884 T^{8} + 27792 p^{3} T^{9} - 186 p^{5} T^{10} + 48 p^{8} T^{11} + 321 p^{8} T^{12} + 6 p^{13} T^{14} + p^{16} T^{16} )^{2}$$
11 $$1 - 432 T^{2} + 7714 p T^{4} - 7780896 T^{6} + 22311049 T^{8} + 70438127616 T^{10} + 13160037970 p T^{12} - 2441540782042416 T^{14} + 459119675281580212 T^{16} - 2441540782042416 p^{4} T^{18} + 13160037970 p^{9} T^{20} + 70438127616 p^{12} T^{22} + 22311049 p^{16} T^{24} - 7780896 p^{20} T^{26} + 7714 p^{25} T^{28} - 432 p^{28} T^{30} + p^{32} T^{32}$$
13 $$( 1 - 1056 T^{2} + 510492 T^{4} - 150812640 T^{6} + 30377120006 T^{8} - 150812640 p^{4} T^{10} + 510492 p^{8} T^{12} - 1056 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
17 $$( 1 + 24 T + 1022 T^{2} + 19920 T^{3} + 479985 T^{4} + 519360 p T^{5} + 194168350 T^{6} + 193952472 p T^{7} + 67118725988 T^{8} + 193952472 p^{3} T^{9} + 194168350 p^{4} T^{10} + 519360 p^{7} T^{11} + 479985 p^{8} T^{12} + 19920 p^{10} T^{13} + 1022 p^{12} T^{14} + 24 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
19 $$1 + 1440 T^{2} + 935910 T^{4} + 23102400 p T^{6} + 186482707321 T^{8} + 56682302220480 T^{10} + 7723885307690070 T^{12} - 285644590063174560 T^{14} -$$$$25\!\cdots\!40$$$$T^{16} - 285644590063174560 p^{4} T^{18} + 7723885307690070 p^{8} T^{20} + 56682302220480 p^{12} T^{22} + 186482707321 p^{16} T^{24} + 23102400 p^{21} T^{26} + 935910 p^{24} T^{28} + 1440 p^{28} T^{30} + p^{32} T^{32}$$
23 $$1 - 1648 T^{2} + 1017702 T^{4} - 249312032 T^{6} + 42919392953 T^{8} - 81219467468736 T^{10} + 45669118532829142 T^{12} + 12560582613209435984 T^{14} -$$$$19\!\cdots\!76$$$$T^{16} + 12560582613209435984 p^{4} T^{18} + 45669118532829142 p^{8} T^{20} - 81219467468736 p^{12} T^{22} + 42919392953 p^{16} T^{24} - 249312032 p^{20} T^{26} + 1017702 p^{24} T^{28} - 1648 p^{28} T^{30} + p^{32} T^{32}$$
29 $$( 1 + 4 T + 2608 T^{2} + 8572 T^{3} + 2977102 T^{4} + 8572 p^{2} T^{5} + 2608 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4}$$
31 $$1 + 4944 T^{2} + 12401142 T^{4} + 21706728288 T^{6} + 30278528813833 T^{8} + 35431200182034624 T^{10} + 36428088221729894790 T^{12} +$$$$35\!\cdots\!36$$$$T^{14} +$$$$33\!\cdots\!48$$$$T^{16} +$$$$35\!\cdots\!36$$$$p^{4} T^{18} + 36428088221729894790 p^{8} T^{20} + 35431200182034624 p^{12} T^{22} + 30278528813833 p^{16} T^{24} + 21706728288 p^{20} T^{26} + 12401142 p^{24} T^{28} + 4944 p^{28} T^{30} + p^{32} T^{32}$$
37 $$( 1 - 36 T - 2658 T^{2} + 179832 T^{3} + 2782297 T^{4} - 349680552 T^{5} + 4944965886 T^{6} + 248707896180 T^{7} - 11947732536204 T^{8} + 248707896180 p^{2} T^{9} + 4944965886 p^{4} T^{10} - 349680552 p^{6} T^{11} + 2782297 p^{8} T^{12} + 179832 p^{10} T^{13} - 2658 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
41 $$( 1 - 10816 T^{2} + 53826588 T^{4} - 162742362560 T^{6} + 329722113042758 T^{8} - 162742362560 p^{4} T^{10} + 53826588 p^{8} T^{12} - 10816 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
43 $$( 1 + 5992 T^{2} + 25433596 T^{4} + 70773684952 T^{6} + 153071479203142 T^{8} + 70773684952 p^{4} T^{10} + 25433596 p^{8} T^{12} + 5992 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
47 $$1 + 7200 T^{2} + 21086790 T^{4} + 446537280 p T^{6} - 28679941197479 T^{8} - 70284530432138880 T^{10} +$$$$21\!\cdots\!30$$$$T^{12} +$$$$13\!\cdots\!80$$$$T^{14} +$$$$38\!\cdots\!20$$$$T^{16} +$$$$13\!\cdots\!80$$$$p^{4} T^{18} +$$$$21\!\cdots\!30$$$$p^{8} T^{20} - 70284530432138880 p^{12} T^{22} - 28679941197479 p^{16} T^{24} + 446537280 p^{21} T^{26} + 21086790 p^{24} T^{28} + 7200 p^{28} T^{30} + p^{32} T^{32}$$
53 $$( 1 - 20 T - 6098 T^{2} + 355352 T^{3} + 299365 p T^{4} - 1398217832 T^{5} + 8057716910 T^{6} + 2171403120260 T^{7} - 80650058761388 T^{8} + 2171403120260 p^{2} T^{9} + 8057716910 p^{4} T^{10} - 1398217832 p^{6} T^{11} + 299365 p^{9} T^{12} + 355352 p^{10} T^{13} - 6098 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
59 $$1 + 13056 T^{2} + 66899190 T^{4} + 216438091392 T^{6} + 1041428149652041 T^{8} + 6415617182341109568 T^{10} +$$$$27\!\cdots\!66$$$$T^{12} +$$$$85\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!80$$$$T^{16} +$$$$85\!\cdots\!44$$$$p^{4} T^{18} +$$$$27\!\cdots\!66$$$$p^{8} T^{20} + 6415617182341109568 p^{12} T^{22} + 1041428149652041 p^{16} T^{24} + 216438091392 p^{20} T^{26} + 66899190 p^{24} T^{28} + 13056 p^{28} T^{30} + p^{32} T^{32}$$
61 $$( 1 - 108 T + 15678 T^{2} - 1273320 T^{3} + 112514745 T^{4} - 8281698840 T^{5} + 635104006110 T^{6} - 41916162290292 T^{7} + 2814046950849908 T^{8} - 41916162290292 p^{2} T^{9} + 635104006110 p^{4} T^{10} - 8281698840 p^{6} T^{11} + 112514745 p^{8} T^{12} - 1273320 p^{10} T^{13} + 15678 p^{12} T^{14} - 108 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
67 $$1 - 19248 T^{2} + 171825062 T^{4} - 1105279125792 T^{6} + 6322631625605113 T^{8} - 28716244420307792256 T^{10} +$$$$95\!\cdots\!42$$$$T^{12} -$$$$31\!\cdots\!16$$$$T^{14} +$$$$13\!\cdots\!84$$$$T^{16} -$$$$31\!\cdots\!16$$$$p^{4} T^{18} +$$$$95\!\cdots\!42$$$$p^{8} T^{20} - 28716244420307792256 p^{12} T^{22} + 6322631625605113 p^{16} T^{24} - 1105279125792 p^{20} T^{26} + 171825062 p^{24} T^{28} - 19248 p^{28} T^{30} + p^{32} T^{32}$$
71 $$( 1 + 24232 T^{2} + 314425660 T^{4} + 2642217676312 T^{6} + 15735155452974982 T^{8} + 2642217676312 p^{4} T^{10} + 314425660 p^{8} T^{12} + 24232 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
73 $$( 1 + 156 T + 22270 T^{2} + 2208648 T^{3} + 203639977 T^{4} + 17676576216 T^{5} + 1406019151294 T^{6} + 115719264813828 T^{7} + 8350708700464372 T^{8} + 115719264813828 p^{2} T^{9} + 1406019151294 p^{4} T^{10} + 17676576216 p^{6} T^{11} + 203639977 p^{8} T^{12} + 2208648 p^{10} T^{13} + 22270 p^{12} T^{14} + 156 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
79 $$1 - 34640 T^{2} + 602988550 T^{4} - 7689396100960 T^{6} + 82922855327705561 T^{8} -$$$$77\!\cdots\!00$$$$T^{10} +$$$$63\!\cdots\!50$$$$T^{12} -$$$$46\!\cdots\!20$$$$T^{14} +$$$$30\!\cdots\!00$$$$T^{16} -$$$$46\!\cdots\!20$$$$p^{4} T^{18} +$$$$63\!\cdots\!50$$$$p^{8} T^{20} -$$$$77\!\cdots\!00$$$$p^{12} T^{22} + 82922855327705561 p^{16} T^{24} - 7689396100960 p^{20} T^{26} + 602988550 p^{24} T^{28} - 34640 p^{28} T^{30} + p^{32} T^{32}$$
83 $$( 1 - 35368 T^{2} + 588987420 T^{4} - 6301011254168 T^{6} + 49538235115611782 T^{8} - 6301011254168 p^{4} T^{10} + 588987420 p^{8} T^{12} - 35368 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
89 $$( 1 - 12 T + 8678 T^{2} - 103560 T^{3} + 5715441 T^{4} + 150778776 T^{5} - 475047477962 T^{6} + 26903544978684 T^{7} - 3925857799931356 T^{8} + 26903544978684 p^{2} T^{9} - 475047477962 p^{4} T^{10} + 150778776 p^{6} T^{11} + 5715441 p^{8} T^{12} - 103560 p^{10} T^{13} + 8678 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
97 $$( 1 - 39520 T^{2} + 789394140 T^{4} - 11378905321376 T^{6} + 124539045704549702 T^{8} - 11378905321376 p^{4} T^{10} + 789394140 p^{8} T^{12} - 39520 p^{12} T^{14} + p^{16} T^{16} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}