# Properties

 Label 16-162e8-1.1-c9e8-0-1 Degree $16$ Conductor $4.744\times 10^{17}$ Sign $1$ Analytic cond. $2.34865\times 10^{15}$ Root an. cond. $9.13432$ Motivic weight $9$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 64·2-s + 1.53e3·4-s + 1.96e3·5-s − 4.49e3·7-s − 1.25e5·10-s − 8.78e3·11-s + 1.62e5·13-s + 2.87e5·14-s − 9.83e5·16-s − 1.07e6·17-s − 1.64e4·19-s + 3.02e6·20-s + 5.62e5·22-s + 2.59e6·23-s + 4.90e6·25-s − 1.04e7·26-s − 6.90e6·28-s + 3.23e6·29-s − 3.48e6·31-s + 2.51e7·32-s + 6.88e7·34-s − 8.84e6·35-s − 4.97e6·37-s + 1.05e6·38-s + 3.46e7·41-s + 4.14e7·43-s − 1.34e7·44-s + ⋯
 L(s)  = 1 − 2.82·2-s + 3·4-s + 1.40·5-s − 0.707·7-s − 3.98·10-s − 0.180·11-s + 1.57·13-s + 2.00·14-s − 3.75·16-s − 3.12·17-s − 0.0289·19-s + 4.22·20-s + 0.511·22-s + 1.93·23-s + 2.51·25-s − 4.46·26-s − 2.12·28-s + 0.848·29-s − 0.677·31-s + 4.24·32-s + 8.83·34-s − 0.996·35-s − 0.436·37-s + 0.0818·38-s + 1.91·41-s + 1.84·43-s − 0.542·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{32}$$ Sign: $1$ Analytic conductor: $$2.34865\times 10^{15}$$ Root analytic conductor: $$9.13432$$ Motivic weight: $$9$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [9/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$3.233357554$$ $$L(\frac12)$$ $$\approx$$ $$3.233357554$$ $$L(\frac{11}{2})$$ 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^{4} T + p^{8} T^{2} )^{4}$$
3 $$1$$
good5 $$1 - 1968 T - 1033142 T^{2} + 5161447392 T^{3} - 5179172183639 T^{4} + 1384901732162592 p T^{5} - 65069990362238686 p^{3} T^{6} -$$$$14\!\cdots\!72$$$$p^{3} T^{7} +$$$$94\!\cdots\!56$$$$p^{4} T^{8} -$$$$14\!\cdots\!72$$$$p^{12} T^{9} - 65069990362238686 p^{21} T^{10} + 1384901732162592 p^{28} T^{11} - 5179172183639 p^{36} T^{12} + 5161447392 p^{45} T^{13} - 1033142 p^{54} T^{14} - 1968 p^{63} T^{15} + p^{72} T^{16}$$
7 $$1 + 4496 T - 134312340 T^{2} - 322254528992 T^{3} + 12375797650597322 T^{4} + 2317622011382359056 p T^{5} -$$$$15\!\cdots\!96$$$$p^{2} T^{6} -$$$$66\!\cdots\!28$$$$p^{3} T^{7} +$$$$14\!\cdots\!19$$$$p^{4} T^{8} -$$$$66\!\cdots\!28$$$$p^{12} T^{9} -$$$$15\!\cdots\!96$$$$p^{20} T^{10} + 2317622011382359056 p^{28} T^{11} + 12375797650597322 p^{36} T^{12} - 322254528992 p^{45} T^{13} - 134312340 p^{54} T^{14} + 4496 p^{63} T^{15} + p^{72} T^{16}$$
11 $$1 + 8784 T - 101412412 p T^{2} - 21402247486176 T^{3} + 6705736084500736186 T^{4} +$$$$15\!\cdots\!00$$$$T^{5} +$$$$17\!\cdots\!40$$$$p T^{6} -$$$$26\!\cdots\!60$$$$T^{7} -$$$$79\!\cdots\!65$$$$T^{8} -$$$$26\!\cdots\!60$$$$p^{9} T^{9} +$$$$17\!\cdots\!40$$$$p^{19} T^{10} +$$$$15\!\cdots\!00$$$$p^{27} T^{11} + 6705736084500736186 p^{36} T^{12} - 21402247486176 p^{45} T^{13} - 101412412 p^{55} T^{14} + 8784 p^{63} T^{15} + p^{72} T^{16}$$
13 $$1 - 162556 T - 4223324994 T^{2} + 1896098353352872 T^{3} - 85260860756219808295 T^{4} +$$$$94\!\cdots\!40$$$$T^{5} -$$$$14\!\cdots\!42$$$$T^{6} -$$$$92\!\cdots\!12$$$$p T^{7} +$$$$21\!\cdots\!24$$$$p^{2} T^{8} -$$$$92\!\cdots\!12$$$$p^{10} T^{9} -$$$$14\!\cdots\!42$$$$p^{18} T^{10} +$$$$94\!\cdots\!40$$$$p^{27} T^{11} - 85260860756219808295 p^{36} T^{12} + 1896098353352872 p^{45} T^{13} - 4223324994 p^{54} T^{14} - 162556 p^{63} T^{15} + p^{72} T^{16}$$
17 $$( 1 + 538080 T + 323513900798 T^{2} + 114203532371811840 T^{3} +$$$$41\!\cdots\!35$$$$T^{4} + 114203532371811840 p^{9} T^{5} + 323513900798 p^{18} T^{6} + 538080 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
19 $$( 1 + 8224 T + 530682376996 T^{2} + 34455495884405920 T^{3} +$$$$19\!\cdots\!10$$$$T^{4} + 34455495884405920 p^{9} T^{5} + 530682376996 p^{18} T^{6} + 8224 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
23 $$1 - 2594736 T - 1409342638820 T^{2} + 4698748269105556896 T^{3} +$$$$43\!\cdots\!46$$$$p T^{4} -$$$$10\!\cdots\!00$$$$T^{5} -$$$$25\!\cdots\!76$$$$T^{6} +$$$$17\!\cdots\!68$$$$T^{7} +$$$$68\!\cdots\!55$$$$T^{8} +$$$$17\!\cdots\!68$$$$p^{9} T^{9} -$$$$25\!\cdots\!76$$$$p^{18} T^{10} -$$$$10\!\cdots\!00$$$$p^{27} T^{11} +$$$$43\!\cdots\!46$$$$p^{37} T^{12} + 4698748269105556896 p^{45} T^{13} - 1409342638820 p^{54} T^{14} - 2594736 p^{63} T^{15} + p^{72} T^{16}$$
29 $$1 - 3232656 T - 22806359737718 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$15\!\cdots\!45$$$$T^{4} -$$$$23\!\cdots\!28$$$$T^{5} +$$$$50\!\cdots\!70$$$$T^{6} +$$$$18\!\cdots\!32$$$$T^{7} -$$$$12\!\cdots\!64$$$$T^{8} +$$$$18\!\cdots\!32$$$$p^{9} T^{9} +$$$$50\!\cdots\!70$$$$p^{18} T^{10} -$$$$23\!\cdots\!28$$$$p^{27} T^{11} +$$$$15\!\cdots\!45$$$$p^{36} T^{12} +$$$$14\!\cdots\!24$$$$p^{45} T^{13} - 22806359737718 p^{54} T^{14} - 3232656 p^{63} T^{15} + p^{72} T^{16}$$
31 $$1 + 3482576 T - 84666387090492 T^{2} -$$$$18\!\cdots\!08$$$$T^{3} +$$$$47\!\cdots\!94$$$$T^{4} +$$$$60\!\cdots\!48$$$$T^{5} -$$$$18\!\cdots\!84$$$$T^{6} -$$$$61\!\cdots\!08$$$$T^{7} +$$$$55\!\cdots\!47$$$$T^{8} -$$$$61\!\cdots\!08$$$$p^{9} T^{9} -$$$$18\!\cdots\!84$$$$p^{18} T^{10} +$$$$60\!\cdots\!48$$$$p^{27} T^{11} +$$$$47\!\cdots\!94$$$$p^{36} T^{12} -$$$$18\!\cdots\!08$$$$p^{45} T^{13} - 84666387090492 p^{54} T^{14} + 3482576 p^{63} T^{15} + p^{72} T^{16}$$
37 $$( 1 + 2487892 T + 426737747557378 T^{2} +$$$$84\!\cdots\!80$$$$T^{3} +$$$$78\!\cdots\!75$$$$T^{4} +$$$$84\!\cdots\!80$$$$p^{9} T^{5} + 426737747557378 p^{18} T^{6} + 2487892 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
41 $$1 - 34657152 T - 344666917407812 T^{2} +$$$$14\!\cdots\!12$$$$T^{3} +$$$$42\!\cdots\!50$$$$T^{4} -$$$$83\!\cdots\!96$$$$T^{5} -$$$$13\!\cdots\!20$$$$T^{6} +$$$$39\!\cdots\!96$$$$T^{7} +$$$$73\!\cdots\!51$$$$T^{8} +$$$$39\!\cdots\!96$$$$p^{9} T^{9} -$$$$13\!\cdots\!20$$$$p^{18} T^{10} -$$$$83\!\cdots\!96$$$$p^{27} T^{11} +$$$$42\!\cdots\!50$$$$p^{36} T^{12} +$$$$14\!\cdots\!12$$$$p^{45} T^{13} - 344666917407812 p^{54} T^{14} - 34657152 p^{63} T^{15} + p^{72} T^{16}$$
43 $$1 - 41410000 T - 285852612011364 T^{2} +$$$$26\!\cdots\!68$$$$T^{3} +$$$$27\!\cdots\!78$$$$T^{4} -$$$$14\!\cdots\!88$$$$T^{5} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$29\!\cdots\!24$$$$T^{7} +$$$$49\!\cdots\!11$$$$T^{8} +$$$$29\!\cdots\!24$$$$p^{9} T^{9} -$$$$11\!\cdots\!24$$$$p^{18} T^{10} -$$$$14\!\cdots\!88$$$$p^{27} T^{11} +$$$$27\!\cdots\!78$$$$p^{36} T^{12} +$$$$26\!\cdots\!68$$$$p^{45} T^{13} - 285852612011364 p^{54} T^{14} - 41410000 p^{63} T^{15} + p^{72} T^{16}$$
47 $$1 - 40558848 T - 467455196135516 T^{2} -$$$$74\!\cdots\!40$$$$T^{3} +$$$$35\!\cdots\!22$$$$T^{4} +$$$$40\!\cdots\!88$$$$T^{5} +$$$$35\!\cdots\!40$$$$T^{6} -$$$$14\!\cdots\!44$$$$T^{7} -$$$$20\!\cdots\!13$$$$T^{8} -$$$$14\!\cdots\!44$$$$p^{9} T^{9} +$$$$35\!\cdots\!40$$$$p^{18} T^{10} +$$$$40\!\cdots\!88$$$$p^{27} T^{11} +$$$$35\!\cdots\!22$$$$p^{36} T^{12} -$$$$74\!\cdots\!40$$$$p^{45} T^{13} - 467455196135516 p^{54} T^{14} - 40558848 p^{63} T^{15} + p^{72} T^{16}$$
53 $$( 1 + 8421024 T + 3661990464011540 T^{2} -$$$$23\!\cdots\!52$$$$T^{3} +$$$$24\!\cdots\!46$$$$T^{4} -$$$$23\!\cdots\!52$$$$p^{9} T^{5} + 3661990464011540 p^{18} T^{6} + 8421024 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
59 $$1 - 26843328 T - 15233430258833900 T^{2} +$$$$63\!\cdots\!80$$$$T^{3} +$$$$10\!\cdots\!98$$$$T^{4} -$$$$57\!\cdots\!08$$$$T^{5} +$$$$37\!\cdots\!72$$$$T^{6} +$$$$24\!\cdots\!48$$$$T^{7} -$$$$75\!\cdots\!09$$$$T^{8} +$$$$24\!\cdots\!48$$$$p^{9} T^{9} +$$$$37\!\cdots\!72$$$$p^{18} T^{10} -$$$$57\!\cdots\!08$$$$p^{27} T^{11} +$$$$10\!\cdots\!98$$$$p^{36} T^{12} +$$$$63\!\cdots\!80$$$$p^{45} T^{13} - 15233430258833900 p^{54} T^{14} - 26843328 p^{63} T^{15} + p^{72} T^{16}$$
61 $$1 - 111504484 T - 32808147451141410 T^{2} +$$$$27\!\cdots\!12$$$$T^{3} +$$$$82\!\cdots\!01$$$$T^{4} -$$$$45\!\cdots\!92$$$$T^{5} -$$$$13\!\cdots\!54$$$$T^{6} +$$$$19\!\cdots\!24$$$$T^{7} +$$$$18\!\cdots\!20$$$$T^{8} +$$$$19\!\cdots\!24$$$$p^{9} T^{9} -$$$$13\!\cdots\!54$$$$p^{18} T^{10} -$$$$45\!\cdots\!92$$$$p^{27} T^{11} +$$$$82\!\cdots\!01$$$$p^{36} T^{12} +$$$$27\!\cdots\!12$$$$p^{45} T^{13} - 32808147451141410 p^{54} T^{14} - 111504484 p^{63} T^{15} + p^{72} T^{16}$$
67 $$1 - 208064512 T - 25688393210852964 T^{2} +$$$$37\!\cdots\!24$$$$T^{3} +$$$$78\!\cdots\!74$$$$T^{4} -$$$$38\!\cdots\!20$$$$T^{5} -$$$$10\!\cdots\!56$$$$T^{6} +$$$$30\!\cdots\!80$$$$T^{7} -$$$$21\!\cdots\!33$$$$T^{8} +$$$$30\!\cdots\!80$$$$p^{9} T^{9} -$$$$10\!\cdots\!56$$$$p^{18} T^{10} -$$$$38\!\cdots\!20$$$$p^{27} T^{11} +$$$$78\!\cdots\!74$$$$p^{36} T^{12} +$$$$37\!\cdots\!24$$$$p^{45} T^{13} - 25688393210852964 p^{54} T^{14} - 208064512 p^{63} T^{15} + p^{72} T^{16}$$
71 $$( 1 + 356008272 T + 129232988292852932 T^{2} +$$$$22\!\cdots\!60$$$$T^{3} +$$$$62\!\cdots\!14$$$$T^{4} +$$$$22\!\cdots\!60$$$$p^{9} T^{5} + 129232988292852932 p^{18} T^{6} + 356008272 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
73 $$( 1 + 37126108 T + 123395685184922122 T^{2} +$$$$20\!\cdots\!88$$$$T^{3} +$$$$73\!\cdots\!35$$$$T^{4} +$$$$20\!\cdots\!88$$$$p^{9} T^{5} + 123395685184922122 p^{18} T^{6} + 37126108 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
79 $$1 - 645134848 T + 9399830430280620 T^{2} +$$$$10\!\cdots\!80$$$$T^{3} -$$$$16\!\cdots\!02$$$$T^{4} -$$$$11\!\cdots\!48$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6} +$$$$98\!\cdots\!48$$$$T^{7} -$$$$84\!\cdots\!49$$$$T^{8} +$$$$98\!\cdots\!48$$$$p^{9} T^{9} +$$$$40\!\cdots\!32$$$$p^{18} T^{10} -$$$$11\!\cdots\!48$$$$p^{27} T^{11} -$$$$16\!\cdots\!02$$$$p^{36} T^{12} +$$$$10\!\cdots\!80$$$$p^{45} T^{13} + 9399830430280620 p^{54} T^{14} - 645134848 p^{63} T^{15} + p^{72} T^{16}$$
83 $$1 - 1019036256 T + 95168392525582708 T^{2} +$$$$13\!\cdots\!36$$$$T^{3} +$$$$50\!\cdots\!10$$$$T^{4} -$$$$52\!\cdots\!68$$$$T^{5} +$$$$18\!\cdots\!36$$$$T^{6} +$$$$12\!\cdots\!88$$$$T^{7} +$$$$26\!\cdots\!39$$$$T^{8} +$$$$12\!\cdots\!88$$$$p^{9} T^{9} +$$$$18\!\cdots\!36$$$$p^{18} T^{10} -$$$$52\!\cdots\!68$$$$p^{27} T^{11} +$$$$50\!\cdots\!10$$$$p^{36} T^{12} +$$$$13\!\cdots\!36$$$$p^{45} T^{13} + 95168392525582708 p^{54} T^{14} - 1019036256 p^{63} T^{15} + p^{72} T^{16}$$
89 $$( 1 + 1548192768 T + 2232987824828793326 T^{2} +$$$$18\!\cdots\!88$$$$T^{3} +$$$$13\!\cdots\!07$$$$T^{4} +$$$$18\!\cdots\!88$$$$p^{9} T^{5} + 2232987824828793326 p^{18} T^{6} + 1548192768 p^{27} T^{7} + p^{36} T^{8} )^{2}$$
97 $$1 + 1112014568 T + 342623331593914980 T^{2} +$$$$30\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!26$$$$T^{4} +$$$$76\!\cdots\!52$$$$T^{5} +$$$$39\!\cdots\!64$$$$T^{6} +$$$$35\!\cdots\!92$$$$T^{7} +$$$$71\!\cdots\!87$$$$T^{8} +$$$$35\!\cdots\!92$$$$p^{9} T^{9} +$$$$39\!\cdots\!64$$$$p^{18} T^{10} +$$$$76\!\cdots\!52$$$$p^{27} T^{11} +$$$$30\!\cdots\!26$$$$p^{36} T^{12} +$$$$30\!\cdots\!56$$$$p^{45} T^{13} + 342623331593914980 p^{54} T^{14} + 1112014568 p^{63} T^{15} + p^{72} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$