# Properties

 Degree $16$ Conductor $54875873536$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 3·3-s + 4·4-s + 5·5-s + 12·6-s − 7-s + 21·9-s + 20·10-s − 155·11-s + 12·12-s + 7·13-s − 4·14-s + 15·15-s + 161·17-s + 84·18-s − 272·19-s + 20·20-s − 3·21-s − 620·22-s + 628·23-s + 129·25-s + 28·26-s − 59·27-s − 4·28-s + 33·29-s + 60·30-s + 323·31-s + ⋯
 L(s)  = 1 + 1.41·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.0539·7-s + 7/9·9-s + 0.632·10-s − 4.24·11-s + 0.288·12-s + 0.149·13-s − 0.0763·14-s + 0.258·15-s + 2.29·17-s + 1.09·18-s − 3.28·19-s + 0.223·20-s − 0.0311·21-s − 6.00·22-s + 5.69·23-s + 1.03·25-s + 0.211·26-s − 0.420·27-s − 0.0269·28-s + 0.211·29-s + 0.365·30-s + 1.87·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 11^{8}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{22} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 11^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.56935$$ $$L(\frac12)$$ $$\approx$$ $$3.56935$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}$$
11 $$1 + 155 T + 13111 T^{2} + 68645 p T^{3} + 264456 p^{2} T^{4} + 68645 p^{4} T^{5} + 13111 p^{6} T^{6} + 155 p^{9} T^{7} + p^{12} T^{8}$$
good3 $$1 - p T - 4 p T^{2} + 158 T^{3} - 1228 T^{4} + 6473 T^{5} + 959 T^{6} - 225344 T^{7} + 1236232 T^{8} - 225344 p^{3} T^{9} + 959 p^{6} T^{10} + 6473 p^{9} T^{11} - 1228 p^{12} T^{12} + 158 p^{15} T^{13} - 4 p^{19} T^{14} - p^{22} T^{15} + p^{24} T^{16}$$
5 $$1 - p T - 104 T^{2} - 326 p T^{3} + 10006 T^{4} + 56777 p T^{5} + 2835571 T^{6} - 1265534 p^{2} T^{7} - 387953844 T^{8} - 1265534 p^{5} T^{9} + 2835571 p^{6} T^{10} + 56777 p^{10} T^{11} + 10006 p^{12} T^{12} - 326 p^{16} T^{13} - 104 p^{18} T^{14} - p^{22} T^{15} + p^{24} T^{16}$$
7 $$1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 211635072 p^{3} T^{9} + 28062801 p^{6} T^{10} - 922401 p^{9} T^{11} - 87888 p^{12} T^{12} - 1104 p^{15} T^{13} + 12 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16}$$
13 $$1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 769971197 p T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 1235524128730 p^{3} T^{9} + 769971197 p^{7} T^{10} + 326819897 p^{9} T^{11} + 10224596 p^{12} T^{12} + 79384 p^{15} T^{13} + 1930 p^{18} T^{14} - 7 p^{21} T^{15} + p^{24} T^{16}$$
17 $$1 - 161 T + 5020 T^{2} + 211188 T^{3} + 21130156 T^{4} - 1323457999 T^{5} - 265862283281 T^{6} + 778694329080 p T^{7} + 277440196391544 T^{8} + 778694329080 p^{4} T^{9} - 265862283281 p^{6} T^{10} - 1323457999 p^{9} T^{11} + 21130156 p^{12} T^{12} + 211188 p^{15} T^{13} + 5020 p^{18} T^{14} - 161 p^{21} T^{15} + p^{24} T^{16}$$
19 $$1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 32108296725040 p^{3} T^{9} + 909384182595 p^{6} T^{10} + 21884364992 p^{9} T^{11} + 292957015 p^{12} T^{12} + 3039584 p^{15} T^{13} + 33201 p^{18} T^{14} + 272 p^{21} T^{15} + p^{24} T^{16}$$
23 $$( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} - 314 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
29 $$1 - 33 T + 14176 T^{2} + 2091144 T^{3} + 419909220 T^{4} + 81926257017 T^{5} + 7198320043835 T^{6} + 2719607629962300 T^{7} + 63828590693142176 T^{8} + 2719607629962300 p^{3} T^{9} + 7198320043835 p^{6} T^{10} + 81926257017 p^{9} T^{11} + 419909220 p^{12} T^{12} + 2091144 p^{15} T^{13} + 14176 p^{18} T^{14} - 33 p^{21} T^{15} + p^{24} T^{16}$$
31 $$1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 1165032156515 p T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - 3925905573337100 p^{3} T^{9} + 1165032156515 p^{7} T^{10} - 93424183073 p^{9} T^{11} + 305184598 p^{12} T^{12} + 5799192 p^{15} T^{13} - 4098 p^{18} T^{14} - 323 p^{21} T^{15} + p^{24} T^{16}$$
37 $$1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - 14416097444880138 p^{3} T^{9} + 91349107260671 p^{6} T^{10} + 1042572020189 p^{9} T^{11} - 566286138 p^{12} T^{12} + 4260746 p^{15} T^{13} - 7868 p^{18} T^{14} - 49 p^{21} T^{15} + p^{24} T^{16}$$
41 $$1 - 361 T - 19140 T^{2} + 12862580 T^{3} + 3853225260 T^{4} - 605378813703 T^{5} - 283363657590617 T^{6} + 99783171614693080 T^{7} - 22290482644766813160 T^{8} + 99783171614693080 p^{3} T^{9} - 283363657590617 p^{6} T^{10} - 605378813703 p^{9} T^{11} + 3853225260 p^{12} T^{12} + 12862580 p^{15} T^{13} - 19140 p^{18} T^{14} - 361 p^{21} T^{15} + p^{24} T^{16}$$
43 $$( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} - 721 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
47 $$1 + 1069 T + 301070 T^{2} - 43368042 T^{3} - 24642745634 T^{4} + 6223420821981 T^{5} + 4635059307315839 T^{6} + 746453682627128500 T^{7} + 29698413240018564184 T^{8} + 746453682627128500 p^{3} T^{9} + 4635059307315839 p^{6} T^{10} + 6223420821981 p^{9} T^{11} - 24642745634 p^{12} T^{12} - 43368042 p^{15} T^{13} + 301070 p^{18} T^{14} + 1069 p^{21} T^{15} + p^{24} T^{16}$$
53 $$1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} -$$$$25\!\cdots\!56$$$$T^{8} + 107096647673733916 p^{3} T^{9} + 8874412003121445 p^{6} T^{10} + 4932564267181 p^{9} T^{11} + 33213391926 p^{12} T^{12} - 60864958 p^{15} T^{13} - 343802 p^{18} T^{14} + 281 p^{21} T^{15} + p^{24} T^{16}$$
59 $$1 + 128 T - 87119 T^{2} - 50674984 T^{3} + 39669673335 T^{4} + 496570131448 T^{5} - 4049042036547885 T^{6} - 22790010341236160 T^{7} +$$$$27\!\cdots\!36$$$$T^{8} - 22790010341236160 p^{3} T^{9} - 4049042036547885 p^{6} T^{10} + 496570131448 p^{9} T^{11} + 39669673335 p^{12} T^{12} - 50674984 p^{15} T^{13} - 87119 p^{18} T^{14} + 128 p^{21} T^{15} + p^{24} T^{16}$$
61 $$1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 1960695737 p^{2} T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} +$$$$68\!\cdots\!36$$$$T^{8} + 3429112403997460700 p^{3} T^{9} - 13214872371636415 p^{6} T^{10} + 1960695737 p^{11} T^{11} + 22253617738 p^{12} T^{12} - 112995618 p^{15} T^{13} - 60798 p^{18} T^{14} + 617 p^{21} T^{15} + p^{24} T^{16}$$
67 $$( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
71 $$1 - 115 T - 138428 T^{2} + 42960000 T^{3} + 77925723838 T^{4} - 71653925274035 T^{5} + 12191505393571699 T^{6} + 3285417296443548450 T^{7} +$$$$11\!\cdots\!80$$$$T^{8} + 3285417296443548450 p^{3} T^{9} + 12191505393571699 p^{6} T^{10} - 71653925274035 p^{9} T^{11} + 77925723838 p^{12} T^{12} + 42960000 p^{15} T^{13} - 138428 p^{18} T^{14} - 115 p^{21} T^{15} + p^{24} T^{16}$$
73 $$1 + 1487 T + 142728 T^{2} - 14316264 p T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} -$$$$19\!\cdots\!28$$$$T^{8} - 68019158899540158384 p^{3} T^{9} + 393291607153115139 p^{6} T^{10} + 366060407928453 p^{9} T^{11} - 569519703768 p^{12} T^{12} - 14316264 p^{16} T^{13} + 142728 p^{18} T^{14} + 1487 p^{21} T^{15} + p^{24} T^{16}$$
79 $$1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} +$$$$59\!\cdots\!76$$$$T^{8} + 8534351363848547400 p^{3} T^{9} - 161731343227399015 p^{6} T^{10} + 97550067497959 p^{9} T^{11} + 350790961998 p^{12} T^{12} - 73013624 p^{15} T^{13} - 500222 p^{18} T^{14} - 71 p^{21} T^{15} + p^{24} T^{16}$$
83 $$1 - 1942 T + 875395 T^{2} + 857035884 T^{3} - 790010610449 T^{4} - 618864428088728 T^{5} + 845503237658086801 T^{6} +$$$$17\!\cdots\!30$$$$T^{7} -$$$$59\!\cdots\!96$$$$T^{8} +$$$$17\!\cdots\!30$$$$p^{3} T^{9} + 845503237658086801 p^{6} T^{10} - 618864428088728 p^{9} T^{11} - 790010610449 p^{12} T^{12} + 857035884 p^{15} T^{13} + 875395 p^{18} T^{14} - 1942 p^{21} T^{15} + p^{24} T^{16}$$
89 $$( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
97 $$1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} +$$$$96\!\cdots\!12$$$$T^{7} +$$$$15\!\cdots\!64$$$$T^{8} +$$$$96\!\cdots\!12$$$$p^{3} T^{9} - 1475957592224572125 p^{6} T^{10} - 4959875190278012 p^{9} T^{11} + 1479496697751 p^{12} T^{12} + 9500759764 p^{15} T^{13} + 10343177 p^{18} T^{14} + 5128 p^{21} T^{15} + p^{24} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$