# Properties

 Label 16-14e8-1.1-c13e8-0-0 Degree $16$ Conductor $1475789056$ Sign $1$ Analytic cond. $2.57979\times 10^{9}$ Root an. cond. $3.87457$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 256·2-s + 182·3-s + 2.45e4·4-s + 1.79e3·5-s + 4.65e4·6-s − 2.68e5·7-s + 3.50e6·9-s + 4.58e5·10-s − 8.72e6·11-s + 4.47e6·12-s + 1.43e6·13-s − 6.86e7·14-s + 3.26e5·15-s − 2.51e8·16-s − 7.94e6·17-s + 8.97e8·18-s − 2.15e8·19-s + 4.40e7·20-s − 4.88e7·21-s − 2.23e9·22-s − 6.19e7·23-s + 1.77e9·25-s + 3.68e8·26-s − 2.60e8·27-s − 6.59e9·28-s − 6.32e9·29-s + 8.34e7·30-s + ⋯
 L(s)  = 1 + 2.82·2-s + 0.144·3-s + 3·4-s + 0.0512·5-s + 0.407·6-s − 0.862·7-s + 2.19·9-s + 0.145·10-s − 1.48·11-s + 0.432·12-s + 0.0826·13-s − 2.43·14-s + 0.00739·15-s − 3.75·16-s − 0.0798·17-s + 6.21·18-s − 1.05·19-s + 0.153·20-s − 0.124·21-s − 4.20·22-s − 0.0872·23-s + 1.45·25-s + 0.233·26-s − 0.129·27-s − 2.58·28-s − 1.97·29-s + 0.0209·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$2.57979\times 10^{9}$$ Root analytic conductor: $$3.87457$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{14} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [13/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(7)$$ $$\approx$$ $$0.2344700904$$ $$L(\frac12)$$ $$\approx$$ $$0.2344700904$$ $$L(\frac{15}{2})$$ 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^{6} T + p^{12} T^{2} )^{4}$$
7 $$1 + 38336 p T - 115258988 p^{3} T^{2} - 1173216704 p^{8} T^{3} + 5981537894 p^{14} T^{4} - 1173216704 p^{21} T^{5} - 115258988 p^{29} T^{6} + 38336 p^{40} T^{7} + p^{52} T^{8}$$
good3 $$1 - 182 T - 3472004 T^{2} + 510005888 p T^{3} + 610111576351 p^{2} T^{4} - 38594331621020 p^{4} T^{5} - 6098702794604744 p^{6} T^{6} + 132316912065600514 p^{9} T^{7} + 7966403831993958904 p^{12} T^{8} + 132316912065600514 p^{22} T^{9} - 6098702794604744 p^{32} T^{10} - 38594331621020 p^{43} T^{11} + 610111576351 p^{54} T^{12} + 510005888 p^{66} T^{13} - 3472004 p^{78} T^{14} - 182 p^{91} T^{15} + p^{104} T^{16}$$
5 $$1 - 1792 T - 1775878222 T^{2} + 184747900060528 T^{3} + 2296434795996497581 T^{4} -$$$$55\!\cdots\!84$$$$p T^{5} +$$$$50\!\cdots\!26$$$$p^{2} T^{6} +$$$$53\!\cdots\!12$$$$p^{4} T^{7} -$$$$11\!\cdots\!96$$$$p^{6} T^{8} +$$$$53\!\cdots\!12$$$$p^{17} T^{9} +$$$$50\!\cdots\!26$$$$p^{28} T^{10} -$$$$55\!\cdots\!84$$$$p^{40} T^{11} + 2296434795996497581 p^{52} T^{12} + 184747900060528 p^{65} T^{13} - 1775878222 p^{78} T^{14} - 1792 p^{91} T^{15} + p^{104} T^{16}$$
11 $$1 + 8726914 T - 25911547427740 T^{2} - 67039793147424510224 p T^{3} -$$$$11\!\cdots\!25$$$$p^{2} T^{4} +$$$$22\!\cdots\!28$$$$p^{3} T^{5} +$$$$12\!\cdots\!92$$$$p^{4} T^{6} -$$$$31\!\cdots\!90$$$$p^{5} T^{7} -$$$$47\!\cdots\!56$$$$p^{6} T^{8} -$$$$31\!\cdots\!90$$$$p^{18} T^{9} +$$$$12\!\cdots\!92$$$$p^{30} T^{10} +$$$$22\!\cdots\!28$$$$p^{42} T^{11} -$$$$11\!\cdots\!25$$$$p^{54} T^{12} - 67039793147424510224 p^{66} T^{13} - 25911547427740 p^{78} T^{14} + 8726914 p^{91} T^{15} + p^{104} T^{16}$$
13 $$( 1 - 719208 T + 661703400953692 T^{2} -$$$$30\!\cdots\!92$$$$T^{3} +$$$$24\!\cdots\!34$$$$T^{4} -$$$$30\!\cdots\!92$$$$p^{13} T^{5} + 661703400953692 p^{26} T^{6} - 719208 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
17 $$1 + 7943068 T - 12420153001480906 T^{2} +$$$$15\!\cdots\!36$$$$T^{3} +$$$$32\!\cdots\!53$$$$T^{4} -$$$$18\!\cdots\!12$$$$T^{5} +$$$$14\!\cdots\!82$$$$T^{6} +$$$$10\!\cdots\!96$$$$T^{7} -$$$$13\!\cdots\!44$$$$T^{8} +$$$$10\!\cdots\!96$$$$p^{13} T^{9} +$$$$14\!\cdots\!82$$$$p^{26} T^{10} -$$$$18\!\cdots\!12$$$$p^{39} T^{11} +$$$$32\!\cdots\!53$$$$p^{52} T^{12} +$$$$15\!\cdots\!36$$$$p^{65} T^{13} - 12420153001480906 p^{78} T^{14} + 7943068 p^{91} T^{15} + p^{104} T^{16}$$
19 $$1 + 215706806 T - 83649252740873580 T^{2} -$$$$64\!\cdots\!84$$$$T^{3} +$$$$61\!\cdots\!95$$$$T^{4} -$$$$73\!\cdots\!52$$$$T^{5} -$$$$31\!\cdots\!04$$$$T^{6} -$$$$14\!\cdots\!30$$$$T^{7} +$$$$11\!\cdots\!32$$$$T^{8} -$$$$14\!\cdots\!30$$$$p^{13} T^{9} -$$$$31\!\cdots\!04$$$$p^{26} T^{10} -$$$$73\!\cdots\!52$$$$p^{39} T^{11} +$$$$61\!\cdots\!95$$$$p^{52} T^{12} -$$$$64\!\cdots\!84$$$$p^{65} T^{13} - 83649252740873580 p^{78} T^{14} + 215706806 p^{91} T^{15} + p^{104} T^{16}$$
23 $$1 + 61927978 T - 1190855939592873856 T^{2} +$$$$30\!\cdots\!76$$$$T^{3} +$$$$72\!\cdots\!19$$$$T^{4} -$$$$28\!\cdots\!84$$$$T^{5} -$$$$22\!\cdots\!24$$$$T^{6} +$$$$84\!\cdots\!66$$$$T^{7} +$$$$67\!\cdots\!48$$$$T^{8} +$$$$84\!\cdots\!66$$$$p^{13} T^{9} -$$$$22\!\cdots\!24$$$$p^{26} T^{10} -$$$$28\!\cdots\!84$$$$p^{39} T^{11} +$$$$72\!\cdots\!19$$$$p^{52} T^{12} +$$$$30\!\cdots\!76$$$$p^{65} T^{13} - 1190855939592873856 p^{78} T^{14} + 61927978 p^{91} T^{15} + p^{104} T^{16}$$
29 $$( 1 + 3162923032 T + 22916063388720489116 T^{2} +$$$$10\!\cdots\!52$$$$T^{3} +$$$$16\!\cdots\!10$$$$T^{4} +$$$$10\!\cdots\!52$$$$p^{13} T^{5} + 22916063388720489116 p^{26} T^{6} + 3162923032 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
31 $$1 - 6113775570 T - 34184846727810893416 T^{2} -$$$$15\!\cdots\!56$$$$T^{3} +$$$$19\!\cdots\!11$$$$T^{4} +$$$$31\!\cdots\!76$$$$T^{5} -$$$$38\!\cdots\!04$$$$T^{6} -$$$$15\!\cdots\!66$$$$T^{7} +$$$$24\!\cdots\!44$$$$T^{8} -$$$$15\!\cdots\!66$$$$p^{13} T^{9} -$$$$38\!\cdots\!04$$$$p^{26} T^{10} +$$$$31\!\cdots\!76$$$$p^{39} T^{11} +$$$$19\!\cdots\!11$$$$p^{52} T^{12} -$$$$15\!\cdots\!56$$$$p^{65} T^{13} - 34184846727810893416 p^{78} T^{14} - 6113775570 p^{91} T^{15} + p^{104} T^{16}$$
37 $$1 + 3945652880 T -$$$$35\!\cdots\!22$$$$T^{2} -$$$$75\!\cdots\!84$$$$T^{3} +$$$$29\!\cdots\!37$$$$T^{4} +$$$$20\!\cdots\!48$$$$T^{5} +$$$$26\!\cdots\!46$$$$T^{6} -$$$$26\!\cdots\!52$$$$T^{7} -$$$$78\!\cdots\!16$$$$T^{8} -$$$$26\!\cdots\!52$$$$p^{13} T^{9} +$$$$26\!\cdots\!46$$$$p^{26} T^{10} +$$$$20\!\cdots\!48$$$$p^{39} T^{11} +$$$$29\!\cdots\!37$$$$p^{52} T^{12} -$$$$75\!\cdots\!84$$$$p^{65} T^{13} -$$$$35\!\cdots\!22$$$$p^{78} T^{14} + 3945652880 p^{91} T^{15} + p^{104} T^{16}$$
41 $$( 1 - 43189289976 T +$$$$25\!\cdots\!84$$$$T^{2} -$$$$40\!\cdots\!84$$$$T^{3} +$$$$19\!\cdots\!30$$$$T^{4} -$$$$40\!\cdots\!84$$$$p^{13} T^{5} +$$$$25\!\cdots\!84$$$$p^{26} T^{6} - 43189289976 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
43 $$( 1 + 54537062128 T +$$$$53\!\cdots\!52$$$$T^{2} +$$$$27\!\cdots\!32$$$$T^{3} +$$$$12\!\cdots\!74$$$$T^{4} +$$$$27\!\cdots\!32$$$$p^{13} T^{5} +$$$$53\!\cdots\!52$$$$p^{26} T^{6} + 54537062128 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
47 $$1 + 3141202722 T -$$$$99\!\cdots\!08$$$$T^{2} +$$$$96\!\cdots\!24$$$$T^{3} +$$$$56\!\cdots\!43$$$$T^{4} -$$$$74\!\cdots\!24$$$$T^{5} +$$$$37\!\cdots\!04$$$$T^{6} +$$$$31\!\cdots\!22$$$$T^{7} -$$$$30\!\cdots\!12$$$$T^{8} +$$$$31\!\cdots\!22$$$$p^{13} T^{9} +$$$$37\!\cdots\!04$$$$p^{26} T^{10} -$$$$74\!\cdots\!24$$$$p^{39} T^{11} +$$$$56\!\cdots\!43$$$$p^{52} T^{12} +$$$$96\!\cdots\!24$$$$p^{65} T^{13} -$$$$99\!\cdots\!08$$$$p^{78} T^{14} + 3141202722 p^{91} T^{15} + p^{104} T^{16}$$
53 $$1 - 149625680376 T -$$$$32\!\cdots\!14$$$$T^{2} +$$$$30\!\cdots\!72$$$$T^{3} +$$$$35\!\cdots\!73$$$$T^{4} +$$$$83\!\cdots\!64$$$$T^{5} -$$$$14\!\cdots\!22$$$$T^{6} -$$$$22\!\cdots\!88$$$$T^{7} +$$$$70\!\cdots\!56$$$$T^{8} -$$$$22\!\cdots\!88$$$$p^{13} T^{9} -$$$$14\!\cdots\!22$$$$p^{26} T^{10} +$$$$83\!\cdots\!64$$$$p^{39} T^{11} +$$$$35\!\cdots\!73$$$$p^{52} T^{12} +$$$$30\!\cdots\!72$$$$p^{65} T^{13} -$$$$32\!\cdots\!14$$$$p^{78} T^{14} - 149625680376 p^{91} T^{15} + p^{104} T^{16}$$
59 $$1 + 866297313938 T +$$$$26\!\cdots\!16$$$$T^{2} +$$$$32\!\cdots\!72$$$$T^{3} -$$$$16\!\cdots\!01$$$$T^{4} -$$$$55\!\cdots\!40$$$$T^{5} -$$$$29\!\cdots\!16$$$$T^{6} -$$$$69\!\cdots\!46$$$$T^{7} -$$$$13\!\cdots\!88$$$$T^{8} -$$$$69\!\cdots\!46$$$$p^{13} T^{9} -$$$$29\!\cdots\!16$$$$p^{26} T^{10} -$$$$55\!\cdots\!40$$$$p^{39} T^{11} -$$$$16\!\cdots\!01$$$$p^{52} T^{12} +$$$$32\!\cdots\!72$$$$p^{65} T^{13} +$$$$26\!\cdots\!16$$$$p^{78} T^{14} + 866297313938 p^{91} T^{15} + p^{104} T^{16}$$
61 $$1 + 477908594184 T -$$$$49\!\cdots\!54$$$$T^{2} -$$$$12\!\cdots\!32$$$$T^{3} +$$$$22\!\cdots\!57$$$$T^{4} +$$$$35\!\cdots\!84$$$$T^{5} -$$$$56\!\cdots\!18$$$$T^{6} -$$$$12\!\cdots\!52$$$$T^{7} +$$$$11\!\cdots\!56$$$$T^{8} -$$$$12\!\cdots\!52$$$$p^{13} T^{9} -$$$$56\!\cdots\!18$$$$p^{26} T^{10} +$$$$35\!\cdots\!84$$$$p^{39} T^{11} +$$$$22\!\cdots\!57$$$$p^{52} T^{12} -$$$$12\!\cdots\!32$$$$p^{65} T^{13} -$$$$49\!\cdots\!54$$$$p^{78} T^{14} + 477908594184 p^{91} T^{15} + p^{104} T^{16}$$
67 $$1 - 1895501016278 T +$$$$16\!\cdots\!44$$$$T^{2} +$$$$36\!\cdots\!92$$$$T^{3} +$$$$20\!\cdots\!99$$$$T^{4} -$$$$15\!\cdots\!64$$$$T^{5} -$$$$63\!\cdots\!80$$$$T^{6} -$$$$28\!\cdots\!02$$$$T^{7} +$$$$12\!\cdots\!00$$$$T^{8} -$$$$28\!\cdots\!02$$$$p^{13} T^{9} -$$$$63\!\cdots\!80$$$$p^{26} T^{10} -$$$$15\!\cdots\!64$$$$p^{39} T^{11} +$$$$20\!\cdots\!99$$$$p^{52} T^{12} +$$$$36\!\cdots\!92$$$$p^{65} T^{13} +$$$$16\!\cdots\!44$$$$p^{78} T^{14} - 1895501016278 p^{91} T^{15} + p^{104} T^{16}$$
71 $$( 1 - 319416336064 T +$$$$11\!\cdots\!92$$$$T^{2} +$$$$18\!\cdots\!52$$$$T^{3} +$$$$28\!\cdots\!78$$$$p T^{4} +$$$$18\!\cdots\!52$$$$p^{13} T^{5} +$$$$11\!\cdots\!92$$$$p^{26} T^{6} - 319416336064 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
73 $$1 + 2966596192756 T -$$$$63\!\cdots\!50$$$$T^{2} -$$$$36\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!25$$$$T^{4} +$$$$21\!\cdots\!56$$$$T^{5} -$$$$21\!\cdots\!70$$$$T^{6} +$$$$68\!\cdots\!04$$$$p T^{7} +$$$$16\!\cdots\!24$$$$p^{2} T^{8} +$$$$68\!\cdots\!04$$$$p^{14} T^{9} -$$$$21\!\cdots\!70$$$$p^{26} T^{10} +$$$$21\!\cdots\!56$$$$p^{39} T^{11} +$$$$16\!\cdots\!25$$$$p^{52} T^{12} -$$$$36\!\cdots\!60$$$$p^{65} T^{13} -$$$$63\!\cdots\!50$$$$p^{78} T^{14} + 2966596192756 p^{91} T^{15} + p^{104} T^{16}$$
79 $$1 + 6505959677634 T +$$$$10\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!68$$$$T^{3} +$$$$13\!\cdots\!83$$$$T^{4} +$$$$31\!\cdots\!32$$$$T^{5} -$$$$17\!\cdots\!68$$$$T^{6} +$$$$60\!\cdots\!74$$$$T^{7} +$$$$43\!\cdots\!56$$$$T^{8} +$$$$60\!\cdots\!74$$$$p^{13} T^{9} -$$$$17\!\cdots\!68$$$$p^{26} T^{10} +$$$$31\!\cdots\!32$$$$p^{39} T^{11} +$$$$13\!\cdots\!83$$$$p^{52} T^{12} +$$$$11\!\cdots\!68$$$$p^{65} T^{13} +$$$$10\!\cdots\!40$$$$p^{78} T^{14} + 6505959677634 p^{91} T^{15} + p^{104} T^{16}$$
83 $$( 1 + 1689908567984 T +$$$$20\!\cdots\!40$$$$T^{2} +$$$$40\!\cdots\!32$$$$T^{3} +$$$$20\!\cdots\!38$$$$T^{4} +$$$$40\!\cdots\!32$$$$p^{13} T^{5} +$$$$20\!\cdots\!40$$$$p^{26} T^{6} + 1689908567984 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
89 $$1 - 9586601667468 T -$$$$11\!\cdots\!14$$$$T^{2} +$$$$13\!\cdots\!04$$$$T^{3} +$$$$18\!\cdots\!49$$$$T^{4} -$$$$53\!\cdots\!00$$$$T^{5} -$$$$54\!\cdots\!90$$$$T^{6} +$$$$55\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!52$$$$T^{8} +$$$$55\!\cdots\!68$$$$p^{13} T^{9} -$$$$54\!\cdots\!90$$$$p^{26} T^{10} -$$$$53\!\cdots\!00$$$$p^{39} T^{11} +$$$$18\!\cdots\!49$$$$p^{52} T^{12} +$$$$13\!\cdots\!04$$$$p^{65} T^{13} -$$$$11\!\cdots\!14$$$$p^{78} T^{14} - 9586601667468 p^{91} T^{15} + p^{104} T^{16}$$
97 $$( 1 + 22280367655784 T +$$$$39\!\cdots\!28$$$$T^{2} +$$$$43\!\cdots\!36$$$$T^{3} +$$$$42\!\cdots\!74$$$$T^{4} +$$$$43\!\cdots\!36$$$$p^{13} T^{5} +$$$$39\!\cdots\!28$$$$p^{26} T^{6} + 22280367655784 p^{39} T^{7} + p^{52} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$