# Properties

 Label 16-162e8-1.1-c10e8-0-0 Degree $16$ Conductor $4.744\times 10^{17}$ Sign $1$ Analytic cond. $1.25969\times 10^{16}$ Root an. cond. $10.1453$ Motivic weight $10$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2.04e3·4-s + 4.51e4·7-s − 7.21e5·13-s + 2.62e6·16-s − 7.31e5·19-s + 5.28e7·25-s − 9.23e7·28-s + 7.25e7·31-s − 1.66e8·37-s − 1.25e7·43-s − 1.83e7·49-s + 1.47e9·52-s − 1.78e9·61-s − 2.68e9·64-s + 1.08e8·67-s + 5.10e9·73-s + 1.49e9·76-s − 9.24e9·79-s − 3.25e10·91-s + 2.24e9·97-s − 1.08e11·100-s + 6.13e10·103-s − 1.17e11·109-s + 1.18e11·112-s + 1.75e11·121-s − 1.48e11·124-s + 127-s + ⋯
 L(s)  = 1 − 2·4-s + 2.68·7-s − 1.94·13-s + 5/2·16-s − 0.295·19-s + 5.41·25-s − 5.36·28-s + 2.53·31-s − 2.40·37-s − 0.0851·43-s − 0.0649·49-s + 3.88·52-s − 2.11·61-s − 5/2·64-s + 0.0805·67-s + 2.46·73-s + 0.590·76-s − 3.00·79-s − 5.21·91-s + 0.261·97-s − 10.8·100-s + 5.29·103-s − 7.64·109-s + 6.71·112-s + 6.77·121-s − 5.06·124-s − 0.792·133-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(11-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{32}$$ Sign: $1$ Analytic conductor: $$1.25969\times 10^{16}$$ Root analytic conductor: $$10.1453$$ Motivic weight: $$10$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [5]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.0002989952097$$ $$L(\frac12)$$ $$\approx$$ $$0.0002989952097$$ $$L(6)$$ 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^{9} T^{2} )^{4}$$
3 $$1$$
good5 $$1 - 10571104 p T^{2} + 55030668906286 p^{2} T^{4} -$$$$18\!\cdots\!64$$$$p^{3} T^{6} +$$$$42\!\cdots\!31$$$$p^{4} T^{8} -$$$$18\!\cdots\!64$$$$p^{23} T^{10} + 55030668906286 p^{42} T^{12} - 10571104 p^{61} T^{14} + p^{80} T^{16}$$
7 $$( 1 - 22556 T + 772331668 T^{2} - 1917011915108 p T^{3} + 6643521062567722 p^{2} T^{4} - 1917011915108 p^{11} T^{5} + 772331668 p^{20} T^{6} - 22556 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
11 $$1 - 175732299920 T^{2} +$$$$14\!\cdots\!64$$$$T^{4} -$$$$68\!\cdots\!60$$$$T^{6} +$$$$21\!\cdots\!26$$$$T^{8} -$$$$68\!\cdots\!60$$$$p^{20} T^{10} +$$$$14\!\cdots\!64$$$$p^{40} T^{12} - 175732299920 p^{60} T^{14} + p^{80} T^{16}$$
13 $$( 1 + 27748 p T + 331408182946 T^{2} + 118099918472653600 T^{3} +$$$$58\!\cdots\!75$$$$T^{4} + 118099918472653600 p^{10} T^{5} + 331408182946 p^{20} T^{6} + 27748 p^{31} T^{7} + p^{40} T^{8} )^{2}$$
17 $$1 - 9754668435248 T^{2} +$$$$47\!\cdots\!34$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$34\!\cdots\!35$$$$T^{8} -$$$$14\!\cdots\!20$$$$p^{20} T^{10} +$$$$47\!\cdots\!34$$$$p^{40} T^{12} - 9754668435248 p^{60} T^{14} + p^{80} T^{16}$$
19 $$( 1 + 365596 T + 10628096140324 T^{2} + 15723599245461508156 T^{3} +$$$$72\!\cdots\!70$$$$T^{4} + 15723599245461508156 p^{10} T^{5} + 10628096140324 p^{20} T^{6} + 365596 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
23 $$1 - 167180527959872 T^{2} +$$$$16\!\cdots\!48$$$$T^{4} -$$$$10\!\cdots\!44$$$$T^{6} +$$$$51\!\cdots\!70$$$$T^{8} -$$$$10\!\cdots\!44$$$$p^{20} T^{10} +$$$$16\!\cdots\!48$$$$p^{40} T^{12} - 167180527959872 p^{60} T^{14} + p^{80} T^{16}$$
29 $$1 - 2322362447160176 T^{2} +$$$$26\!\cdots\!30$$$$T^{4} -$$$$19\!\cdots\!44$$$$T^{6} +$$$$95\!\cdots\!39$$$$T^{8} -$$$$19\!\cdots\!44$$$$p^{20} T^{10} +$$$$26\!\cdots\!30$$$$p^{40} T^{12} - 2322362447160176 p^{60} T^{14} + p^{80} T^{16}$$
31 $$( 1 - 36251264 T + 2335436830550956 T^{2} -$$$$62\!\cdots\!88$$$$T^{3} +$$$$25\!\cdots\!06$$$$T^{4} -$$$$62\!\cdots\!88$$$$p^{10} T^{5} + 2335436830550956 p^{20} T^{6} - 36251264 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
37 $$( 1 + 83476120 T + 15137019401494870 T^{2} +$$$$11\!\cdots\!80$$$$T^{3} +$$$$98\!\cdots\!67$$$$T^{4} +$$$$11\!\cdots\!80$$$$p^{10} T^{5} + 15137019401494870 p^{20} T^{6} + 83476120 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
41 $$1 - 52900132959553136 T^{2} +$$$$13\!\cdots\!84$$$$T^{4} -$$$$25\!\cdots\!16$$$$T^{6} +$$$$37\!\cdots\!90$$$$T^{8} -$$$$25\!\cdots\!16$$$$p^{20} T^{10} +$$$$13\!\cdots\!84$$$$p^{40} T^{12} - 52900132959553136 p^{60} T^{14} + p^{80} T^{16}$$
43 $$( 1 + 6255220 T + 59946472373598052 T^{2} +$$$$52\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!58$$$$T^{4} +$$$$52\!\cdots\!80$$$$p^{10} T^{5} + 59946472373598052 p^{20} T^{6} + 6255220 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
47 $$1 - 305022268387356968 T^{2} +$$$$42\!\cdots\!04$$$$T^{4} -$$$$36\!\cdots\!40$$$$T^{6} +$$$$22\!\cdots\!30$$$$T^{8} -$$$$36\!\cdots\!40$$$$p^{20} T^{10} +$$$$42\!\cdots\!04$$$$p^{40} T^{12} - 305022268387356968 p^{60} T^{14} + p^{80} T^{16}$$
53 $$1 - 842830763760485360 T^{2} +$$$$36\!\cdots\!88$$$$T^{4} -$$$$10\!\cdots\!80$$$$T^{6} +$$$$21\!\cdots\!78$$$$T^{8} -$$$$10\!\cdots\!80$$$$p^{20} T^{10} +$$$$36\!\cdots\!88$$$$p^{40} T^{12} - 842830763760485360 p^{60} T^{14} + p^{80} T^{16}$$
59 $$1 - 2467177976527842344 T^{2} +$$$$31\!\cdots\!16$$$$T^{4} -$$$$25\!\cdots\!28$$$$T^{6} +$$$$15\!\cdots\!86$$$$T^{8} -$$$$25\!\cdots\!28$$$$p^{20} T^{10} +$$$$31\!\cdots\!16$$$$p^{40} T^{12} - 2467177976527842344 p^{60} T^{14} + p^{80} T^{16}$$
61 $$( 1 + 891136624 T + 959003895617693566 T^{2} +$$$$76\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!11$$$$T^{4} +$$$$76\!\cdots\!68$$$$p^{10} T^{5} + 959003895617693566 p^{20} T^{6} + 891136624 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
67 $$( 1 - 54371828 T + 6949081761198061684 T^{2} -$$$$18\!\cdots\!24$$$$T^{3} +$$$$18\!\cdots\!26$$$$T^{4} -$$$$18\!\cdots\!24$$$$p^{10} T^{5} + 6949081761198061684 p^{20} T^{6} - 54371828 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
71 $$1 - 11772016379311621760 T^{2} +$$$$64\!\cdots\!88$$$$T^{4} -$$$$21\!\cdots\!00$$$$T^{6} +$$$$64\!\cdots\!98$$$$T^{8} -$$$$21\!\cdots\!00$$$$p^{20} T^{10} +$$$$64\!\cdots\!88$$$$p^{40} T^{12} - 11772016379311621760 p^{60} T^{14} + p^{80} T^{16}$$
73 $$( 1 - 2551117184 T + 12044333407858330222 T^{2} -$$$$22\!\cdots\!52$$$$T^{3} +$$$$66\!\cdots\!15$$$$T^{4} -$$$$22\!\cdots\!52$$$$p^{10} T^{5} + 12044333407858330222 p^{20} T^{6} - 2551117184 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
79 $$( 1 + 4623100876 T + 21165890035194757924 T^{2} +$$$$72\!\cdots\!16$$$$T^{3} +$$$$24\!\cdots\!30$$$$T^{4} +$$$$72\!\cdots\!16$$$$p^{10} T^{5} + 21165890035194757924 p^{20} T^{6} + 4623100876 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
83 $$1 - 17141683756914722216 T^{2} +$$$$23\!\cdots\!76$$$$T^{4} -$$$$25\!\cdots\!12$$$$T^{6} +$$$$17\!\cdots\!26$$$$T^{8} -$$$$25\!\cdots\!12$$$$p^{20} T^{10} +$$$$23\!\cdots\!76$$$$p^{40} T^{12} - 17141683756914722216 p^{60} T^{14} + p^{80} T^{16}$$
89 $$1 -$$$$16\!\cdots\!12$$$$T^{2} +$$$$14\!\cdots\!54$$$$T^{4} -$$$$74\!\cdots\!20$$$$T^{6} +$$$$27\!\cdots\!95$$$$T^{8} -$$$$74\!\cdots\!20$$$$p^{20} T^{10} +$$$$14\!\cdots\!54$$$$p^{40} T^{12} -$$$$16\!\cdots\!12$$$$p^{60} T^{14} + p^{80} T^{16}$$
97 $$( 1 - 1123887968 T +$$$$21\!\cdots\!40$$$$T^{2} -$$$$44\!\cdots\!88$$$$T^{3} +$$$$20\!\cdots\!54$$$$T^{4} -$$$$44\!\cdots\!88$$$$p^{10} T^{5} +$$$$21\!\cdots\!40$$$$p^{20} T^{6} - 1123887968 p^{30} T^{7} + p^{40} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$