# Properties

 Label 16-63e8-1.1-c19e8-0-0 Degree $16$ Conductor $2.482\times 10^{14}$ Sign $1$ Analytic cond. $1.86477\times 10^{17}$ Root an. cond. $12.0064$ Motivic weight $19$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $8$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 1.03e6·4-s + 3.22e8·7-s − 1.39e11·13-s + 6.88e11·16-s + 2.34e12·19-s − 9.62e13·25-s − 3.34e14·28-s − 4.60e14·31-s − 1.50e15·37-s + 1.24e15·43-s + 5.86e16·49-s + 1.44e17·52-s + 1.12e17·61-s − 2.88e17·64-s − 1.51e17·67-s − 7.36e17·73-s − 2.42e18·76-s − 1.86e18·79-s − 4.51e19·91-s − 9.32e18·97-s + 9.96e19·100-s − 1.51e19·103-s − 3.95e19·109-s + 2.22e20·112-s − 2.49e20·121-s + 4.77e20·124-s + 127-s + ⋯
 L(s)  = 1 − 1.97·4-s + 3.02·7-s − 3.65·13-s + 2.50·16-s + 1.66·19-s − 5.04·25-s − 5.97·28-s − 3.13·31-s − 1.89·37-s + 0.376·43-s + 36/7·49-s + 7.22·52-s + 1.22·61-s − 2.00·64-s − 0.680·67-s − 1.46·73-s − 3.29·76-s − 1.75·79-s − 11.0·91-s − 1.24·97-s + 9.96·100-s − 1.14·103-s − 1.74·109-s + 7.57·112-s − 4.07·121-s + 6.18·124-s + 5.04·133-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+19/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$3^{16} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$1.86477\times 10^{17}$$ Root analytic conductor: $$12.0064$$ Motivic weight: $$19$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$8$$ Selberg data: $$(16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [19/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(10)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{21}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$( 1 - p^{9} T )^{8}$$
good2 $$1 + 258943 p^{2} T^{2} + 1499685103 p^{8} T^{4} - 410605443665 p^{16} T^{6} - 1168378983796931 p^{26} T^{8} - 410605443665 p^{54} T^{10} + 1499685103 p^{84} T^{12} + 258943 p^{116} T^{14} + p^{152} T^{16}$$
5 $$1 + 3848885859256 p^{2} T^{2} +$$$$76\!\cdots\!48$$$$p^{4} T^{4} +$$$$97\!\cdots\!84$$$$p^{6} T^{6} +$$$$88\!\cdots\!26$$$$p^{8} T^{8} +$$$$97\!\cdots\!84$$$$p^{44} T^{10} +$$$$76\!\cdots\!48$$$$p^{80} T^{12} + 3848885859256 p^{116} T^{14} + p^{152} T^{16}$$
11 $$1 +$$$$24\!\cdots\!24$$$$T^{2} +$$$$29\!\cdots\!88$$$$T^{4} +$$$$19\!\cdots\!32$$$$p^{2} T^{6} +$$$$10\!\cdots\!74$$$$p^{4} T^{8} +$$$$19\!\cdots\!32$$$$p^{40} T^{10} +$$$$29\!\cdots\!88$$$$p^{76} T^{12} +$$$$24\!\cdots\!24$$$$p^{114} T^{14} + p^{152} T^{16}$$
13 $$( 1 + 69886183192 T +$$$$52\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!96$$$$p T^{3} +$$$$64\!\cdots\!94$$$$p^{2} T^{4} +$$$$20\!\cdots\!96$$$$p^{20} T^{5} +$$$$52\!\cdots\!00$$$$p^{38} T^{6} + 69886183192 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
17 $$1 +$$$$68\!\cdots\!16$$$$T^{2} +$$$$85\!\cdots\!60$$$$p^{2} T^{4} +$$$$56\!\cdots\!36$$$$p^{5} T^{6} +$$$$92\!\cdots\!82$$$$p^{6} T^{8} +$$$$56\!\cdots\!36$$$$p^{43} T^{10} +$$$$85\!\cdots\!60$$$$p^{78} T^{12} +$$$$68\!\cdots\!16$$$$p^{114} T^{14} + p^{152} T^{16}$$
19 $$( 1 - 1172443419728 T +$$$$50\!\cdots\!52$$$$T^{2} -$$$$49\!\cdots\!96$$$$T^{3} +$$$$14\!\cdots\!34$$$$T^{4} -$$$$49\!\cdots\!96$$$$p^{19} T^{5} +$$$$50\!\cdots\!52$$$$p^{38} T^{6} - 1172443419728 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
23 $$1 +$$$$35\!\cdots\!20$$$$T^{2} +$$$$61\!\cdots\!04$$$$T^{4} +$$$$70\!\cdots\!20$$$$T^{6} +$$$$59\!\cdots\!26$$$$T^{8} +$$$$70\!\cdots\!20$$$$p^{38} T^{10} +$$$$61\!\cdots\!04$$$$p^{76} T^{12} +$$$$35\!\cdots\!20$$$$p^{114} T^{14} + p^{152} T^{16}$$
29 $$1 +$$$$41\!\cdots\!44$$$$T^{2} +$$$$78\!\cdots\!12$$$$T^{4} +$$$$89\!\cdots\!00$$$$T^{6} +$$$$66\!\cdots\!86$$$$T^{8} +$$$$89\!\cdots\!00$$$$p^{38} T^{10} +$$$$78\!\cdots\!12$$$$p^{76} T^{12} +$$$$41\!\cdots\!44$$$$p^{114} T^{14} + p^{152} T^{16}$$
31 $$( 1 + 230486396113456 T +$$$$43\!\cdots\!60$$$$T^{2} +$$$$58\!\cdots\!04$$$$T^{3} +$$$$82\!\cdots\!54$$$$T^{4} +$$$$58\!\cdots\!04$$$$p^{19} T^{5} +$$$$43\!\cdots\!60$$$$p^{38} T^{6} + 230486396113456 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
37 $$( 1 + 750085021566424 T +$$$$15\!\cdots\!08$$$$T^{2} +$$$$11\!\cdots\!20$$$$T^{3} +$$$$12\!\cdots\!06$$$$T^{4} +$$$$11\!\cdots\!20$$$$p^{19} T^{5} +$$$$15\!\cdots\!08$$$$p^{38} T^{6} + 750085021566424 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
41 $$1 +$$$$13\!\cdots\!00$$$$T^{2} +$$$$10\!\cdots\!92$$$$T^{4} +$$$$51\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!98$$$$T^{8} +$$$$51\!\cdots\!00$$$$p^{38} T^{10} +$$$$10\!\cdots\!92$$$$p^{76} T^{12} +$$$$13\!\cdots\!00$$$$p^{114} T^{14} + p^{152} T^{16}$$
43 $$( 1 - 620247319381328 T +$$$$18\!\cdots\!20$$$$T^{2} -$$$$65\!\cdots\!32$$$$T^{3} +$$$$15\!\cdots\!86$$$$T^{4} -$$$$65\!\cdots\!32$$$$p^{19} T^{5} +$$$$18\!\cdots\!20$$$$p^{38} T^{6} - 620247319381328 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
47 $$1 +$$$$29\!\cdots\!88$$$$T^{2} +$$$$39\!\cdots\!12$$$$T^{4} +$$$$31\!\cdots\!76$$$$T^{6} +$$$$20\!\cdots\!94$$$$T^{8} +$$$$31\!\cdots\!76$$$$p^{38} T^{10} +$$$$39\!\cdots\!12$$$$p^{76} T^{12} +$$$$29\!\cdots\!88$$$$p^{114} T^{14} + p^{152} T^{16}$$
53 $$1 +$$$$35\!\cdots\!64$$$$T^{2} +$$$$60\!\cdots\!60$$$$T^{4} +$$$$63\!\cdots\!48$$$$T^{6} +$$$$44\!\cdots\!38$$$$T^{8} +$$$$63\!\cdots\!48$$$$p^{38} T^{10} +$$$$60\!\cdots\!60$$$$p^{76} T^{12} +$$$$35\!\cdots\!64$$$$p^{114} T^{14} + p^{152} T^{16}$$
59 $$1 +$$$$22\!\cdots\!68$$$$T^{2} +$$$$25\!\cdots\!80$$$$T^{4} +$$$$18\!\cdots\!44$$$$T^{6} +$$$$94\!\cdots\!78$$$$T^{8} +$$$$18\!\cdots\!44$$$$p^{38} T^{10} +$$$$25\!\cdots\!80$$$$p^{76} T^{12} +$$$$22\!\cdots\!68$$$$p^{114} T^{14} + p^{152} T^{16}$$
61 $$( 1 - 56128048864837880 T +$$$$12\!\cdots\!64$$$$T^{2} -$$$$32\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!86$$$$T^{4} -$$$$32\!\cdots\!40$$$$p^{19} T^{5} +$$$$12\!\cdots\!64$$$$p^{38} T^{6} - 56128048864837880 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
67 $$( 1 + 75733494970155568 T +$$$$12\!\cdots\!64$$$$T^{2} +$$$$22\!\cdots\!76$$$$T^{3} +$$$$75\!\cdots\!70$$$$T^{4} +$$$$22\!\cdots\!76$$$$p^{19} T^{5} +$$$$12\!\cdots\!64$$$$p^{38} T^{6} + 75733494970155568 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
71 $$1 +$$$$10\!\cdots\!20$$$$T^{2} +$$$$32\!\cdots\!32$$$$T^{4} +$$$$68\!\cdots\!40$$$$T^{6} +$$$$45\!\cdots\!98$$$$T^{8} +$$$$68\!\cdots\!40$$$$p^{38} T^{10} +$$$$32\!\cdots\!32$$$$p^{76} T^{12} +$$$$10\!\cdots\!20$$$$p^{114} T^{14} + p^{152} T^{16}$$
73 $$( 1 + 368041086891678088 T +$$$$69\!\cdots\!80$$$$T^{2} +$$$$11\!\cdots\!92$$$$T^{3} +$$$$20\!\cdots\!86$$$$T^{4} +$$$$11\!\cdots\!92$$$$p^{19} T^{5} +$$$$69\!\cdots\!80$$$$p^{38} T^{6} + 368041086891678088 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
79 $$( 1 + 934021788005299360 T +$$$$27\!\cdots\!08$$$$T^{2} +$$$$22\!\cdots\!80$$$$T^{3} +$$$$40\!\cdots\!38$$$$T^{4} +$$$$22\!\cdots\!80$$$$p^{19} T^{5} +$$$$27\!\cdots\!08$$$$p^{38} T^{6} + 934021788005299360 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
83 $$1 +$$$$80\!\cdots\!88$$$$T^{2} +$$$$53\!\cdots\!72$$$$T^{4} +$$$$21\!\cdots\!76$$$$T^{6} +$$$$76\!\cdots\!34$$$$T^{8} +$$$$21\!\cdots\!76$$$$p^{38} T^{10} +$$$$53\!\cdots\!72$$$$p^{76} T^{12} +$$$$80\!\cdots\!88$$$$p^{114} T^{14} + p^{152} T^{16}$$
89 $$1 +$$$$42\!\cdots\!20$$$$T^{2} +$$$$11\!\cdots\!72$$$$T^{4} +$$$$19\!\cdots\!40$$$$T^{6} +$$$$24\!\cdots\!18$$$$T^{8} +$$$$19\!\cdots\!40$$$$p^{38} T^{10} +$$$$11\!\cdots\!72$$$$p^{76} T^{12} +$$$$42\!\cdots\!20$$$$p^{114} T^{14} + p^{152} T^{16}$$
97 $$( 1 + 4660331432528518600 T +$$$$21\!\cdots\!24$$$$T^{2} +$$$$73\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!22$$$$T^{4} +$$$$73\!\cdots\!00$$$$p^{19} T^{5} +$$$$21\!\cdots\!24$$$$p^{38} T^{6} + 4660331432528518600 p^{57} T^{7} + p^{76} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$