# Properties

 Degree 16 Conductor $5^{8}$ Sign $1$ Motivic weight 17 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2.34e5·4-s + 3.79e5·5-s + 3.99e8·9-s + 4.63e8·11-s + 1.62e10·16-s − 2.06e10·19-s + 8.90e10·20-s − 8.22e11·25-s − 4.07e12·29-s + 1.13e13·31-s + 9.36e13·36-s + 9.72e13·41-s + 1.08e14·44-s + 1.51e14·45-s + 5.02e14·49-s + 1.75e14·55-s − 1.09e15·59-s + 8.06e15·61-s − 4.17e14·64-s + 2.52e16·71-s − 4.83e15·76-s + 3.22e15·79-s + 6.17e15·80-s + 5.97e16·81-s + 9.29e16·89-s − 7.81e15·95-s + 1.84e17·99-s + ⋯
 L(s)  = 1 + 1.79·4-s + 0.434·5-s + 3.09·9-s + 0.651·11-s + 0.948·16-s − 0.278·19-s + 0.777·20-s − 1.07·25-s − 1.51·29-s + 2.38·31-s + 5.53·36-s + 1.90·41-s + 1.16·44-s + 1.34·45-s + 2.15·49-s + 0.282·55-s − 0.967·59-s + 5.38·61-s − 0.185·64-s + 4.63·71-s − 0.498·76-s + 0.239·79-s + 0.411·80-s + 3.58·81-s + 2.50·89-s − 0.120·95-s + 2.01·99-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+17/2)^{8} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$390625$$    =    $$5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$17$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 390625,\ (\ :[17/2]^{8}),\ 1)$ $L(9)$ $\approx$ $35.4432$ $L(\frac12)$ $\approx$ $35.4432$ $L(\frac{19}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 16. If $p = 5$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad5 $$1 - 15168 p^{2} T + 309286732 p^{5} T^{2} + 177748146432 p^{9} T^{3} + 2397588578934 p^{16} T^{4} + 177748146432 p^{26} T^{5} + 309286732 p^{39} T^{6} - 15168 p^{53} T^{7} + p^{68} T^{8}$$
good2 $$1 - 58685 p^{2} T^{2} + 606488299 p^{6} T^{4} - 148653192985 p^{15} T^{6} + 178370738447659 p^{22} T^{8} - 148653192985 p^{49} T^{10} + 606488299 p^{74} T^{12} - 58685 p^{104} T^{14} + p^{136} T^{16}$$
3 $$1 - 44344720 p^{2} T^{2} + 45525406383748 p^{7} T^{4} -$$$$30\!\cdots\!40$$$$p^{10} T^{6} +$$$$65\!\cdots\!94$$$$p^{18} T^{8} -$$$$30\!\cdots\!40$$$$p^{44} T^{10} + 45525406383748 p^{75} T^{12} - 44344720 p^{104} T^{14} + p^{136} T^{16}$$
7 $$1 - 502234877138000 T^{2} +$$$$38\!\cdots\!04$$$$p^{2} T^{4} -$$$$75\!\cdots\!00$$$$p^{7} T^{6} +$$$$57\!\cdots\!94$$$$p^{10} T^{8} -$$$$75\!\cdots\!00$$$$p^{41} T^{10} +$$$$38\!\cdots\!04$$$$p^{70} T^{12} - 502234877138000 p^{102} T^{14} + p^{136} T^{16}$$
11 $$( 1 - 231648288 T + 64273843738603108 p T^{2} +$$$$16\!\cdots\!84$$$$p^{2} T^{3} +$$$$34\!\cdots\!70$$$$p^{3} T^{4} +$$$$16\!\cdots\!84$$$$p^{19} T^{5} + 64273843738603108 p^{35} T^{6} - 231648288 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
13 $$1 - 39268035547036882760 T^{2} +$$$$47\!\cdots\!24$$$$p^{2} T^{4} -$$$$39\!\cdots\!20$$$$p^{4} T^{6} +$$$$23\!\cdots\!14$$$$p^{6} T^{8} -$$$$39\!\cdots\!20$$$$p^{38} T^{10} +$$$$47\!\cdots\!24$$$$p^{70} T^{12} - 39268035547036882760 p^{102} T^{14} + p^{136} T^{16}$$
17 $$1 -$$$$25\!\cdots\!20$$$$T^{2} +$$$$32\!\cdots\!16$$$$T^{4} -$$$$32\!\cdots\!40$$$$T^{6} +$$$$29\!\cdots\!46$$$$T^{8} -$$$$32\!\cdots\!40$$$$p^{34} T^{10} +$$$$32\!\cdots\!16$$$$p^{68} T^{12} -$$$$25\!\cdots\!20$$$$p^{102} T^{14} + p^{136} T^{16}$$
19 $$( 1 + 10307856640 T +$$$$16\!\cdots\!56$$$$T^{2} +$$$$12\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!26$$$$T^{4} +$$$$12\!\cdots\!80$$$$p^{17} T^{5} +$$$$16\!\cdots\!56$$$$p^{34} T^{6} + 10307856640 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
23 $$1 -$$$$60\!\cdots\!40$$$$T^{2} +$$$$19\!\cdots\!36$$$$T^{4} -$$$$43\!\cdots\!80$$$$T^{6} +$$$$71\!\cdots\!86$$$$T^{8} -$$$$43\!\cdots\!80$$$$p^{34} T^{10} +$$$$19\!\cdots\!36$$$$p^{68} T^{12} -$$$$60\!\cdots\!40$$$$p^{102} T^{14} + p^{136} T^{16}$$
29 $$( 1 + 2039508912360 T +$$$$21\!\cdots\!36$$$$T^{2} +$$$$45\!\cdots\!20$$$$T^{3} +$$$$20\!\cdots\!86$$$$T^{4} +$$$$45\!\cdots\!20$$$$p^{17} T^{5} +$$$$21\!\cdots\!36$$$$p^{34} T^{6} + 2039508912360 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
31 $$( 1 - 5664664329248 T +$$$$88\!\cdots\!08$$$$T^{2} -$$$$32\!\cdots\!96$$$$T^{3} +$$$$28\!\cdots\!70$$$$T^{4} -$$$$32\!\cdots\!96$$$$p^{17} T^{5} +$$$$88\!\cdots\!08$$$$p^{34} T^{6} - 5664664329248 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
37 $$1 -$$$$27\!\cdots\!60$$$$T^{2} +$$$$34\!\cdots\!56$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$13\!\cdots\!26$$$$T^{8} -$$$$26\!\cdots\!20$$$$p^{34} T^{10} +$$$$34\!\cdots\!56$$$$p^{68} T^{12} -$$$$27\!\cdots\!60$$$$p^{102} T^{14} + p^{136} T^{16}$$
41 $$( 1 - 48608626423728 T +$$$$55\!\cdots\!68$$$$T^{2} -$$$$40\!\cdots\!76$$$$T^{3} +$$$$10\!\cdots\!70$$$$T^{4} -$$$$40\!\cdots\!76$$$$p^{17} T^{5} +$$$$55\!\cdots\!68$$$$p^{34} T^{6} - 48608626423728 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
43 $$1 -$$$$31\!\cdots\!00$$$$T^{2} +$$$$49\!\cdots\!96$$$$T^{4} -$$$$50\!\cdots\!00$$$$T^{6} +$$$$34\!\cdots\!06$$$$T^{8} -$$$$50\!\cdots\!00$$$$p^{34} T^{10} +$$$$49\!\cdots\!96$$$$p^{68} T^{12} -$$$$31\!\cdots\!00$$$$p^{102} T^{14} + p^{136} T^{16}$$
47 $$1 -$$$$12\!\cdots\!80$$$$T^{2} +$$$$79\!\cdots\!76$$$$T^{4} -$$$$32\!\cdots\!60$$$$T^{6} +$$$$97\!\cdots\!66$$$$T^{8} -$$$$32\!\cdots\!60$$$$p^{34} T^{10} +$$$$79\!\cdots\!76$$$$p^{68} T^{12} -$$$$12\!\cdots\!80$$$$p^{102} T^{14} + p^{136} T^{16}$$
53 $$1 -$$$$11\!\cdots\!80$$$$T^{2} +$$$$59\!\cdots\!76$$$$T^{4} -$$$$19\!\cdots\!60$$$$T^{6} +$$$$47\!\cdots\!66$$$$T^{8} -$$$$19\!\cdots\!60$$$$p^{34} T^{10} +$$$$59\!\cdots\!76$$$$p^{68} T^{12} -$$$$11\!\cdots\!80$$$$p^{102} T^{14} + p^{136} T^{16}$$
59 $$( 1 + 545704756120320 T +$$$$31\!\cdots\!76$$$$T^{2} +$$$$93\!\cdots\!40$$$$T^{3} +$$$$45\!\cdots\!66$$$$T^{4} +$$$$93\!\cdots\!40$$$$p^{17} T^{5} +$$$$31\!\cdots\!76$$$$p^{34} T^{6} + 545704756120320 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
61 $$( 1 - 4032365505387488 T +$$$$10\!\cdots\!88$$$$T^{2} -$$$$18\!\cdots\!36$$$$T^{3} +$$$$30\!\cdots\!70$$$$T^{4} -$$$$18\!\cdots\!36$$$$p^{17} T^{5} +$$$$10\!\cdots\!88$$$$p^{34} T^{6} - 4032365505387488 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
67 $$1 -$$$$56\!\cdots\!20$$$$T^{2} +$$$$16\!\cdots\!16$$$$T^{4} -$$$$30\!\cdots\!40$$$$T^{6} +$$$$39\!\cdots\!46$$$$T^{8} -$$$$30\!\cdots\!40$$$$p^{34} T^{10} +$$$$16\!\cdots\!16$$$$p^{68} T^{12} -$$$$56\!\cdots\!20$$$$p^{102} T^{14} + p^{136} T^{16}$$
71 $$( 1 - 12620746029070368 T +$$$$16\!\cdots\!48$$$$T^{2} -$$$$11\!\cdots\!16$$$$T^{3} +$$$$81\!\cdots\!70$$$$T^{4} -$$$$11\!\cdots\!16$$$$p^{17} T^{5} +$$$$16\!\cdots\!48$$$$p^{34} T^{6} - 12620746029070368 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
73 $$1 -$$$$25\!\cdots\!40$$$$T^{2} +$$$$30\!\cdots\!36$$$$T^{4} -$$$$24\!\cdots\!80$$$$T^{6} +$$$$13\!\cdots\!86$$$$T^{8} -$$$$24\!\cdots\!80$$$$p^{34} T^{10} +$$$$30\!\cdots\!36$$$$p^{68} T^{12} -$$$$25\!\cdots\!40$$$$p^{102} T^{14} + p^{136} T^{16}$$
79 $$( 1 - 20442103403360 p T +$$$$57\!\cdots\!36$$$$T^{2} -$$$$80\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!86$$$$T^{4} -$$$$80\!\cdots\!80$$$$p^{17} T^{5} +$$$$57\!\cdots\!36$$$$p^{34} T^{6} - 20442103403360 p^{52} T^{7} + p^{68} T^{8} )^{2}$$
83 $$1 -$$$$13\!\cdots\!20$$$$T^{2} +$$$$14\!\cdots\!52$$$$p T^{4} -$$$$79\!\cdots\!40$$$$T^{6} +$$$$37\!\cdots\!46$$$$T^{8} -$$$$79\!\cdots\!40$$$$p^{34} T^{10} +$$$$14\!\cdots\!52$$$$p^{69} T^{12} -$$$$13\!\cdots\!20$$$$p^{102} T^{14} + p^{136} T^{16}$$
89 $$( 1 - 46481993767813320 T +$$$$39\!\cdots\!16$$$$T^{2} -$$$$17\!\cdots\!40$$$$T^{3} +$$$$70\!\cdots\!46$$$$T^{4} -$$$$17\!\cdots\!40$$$$p^{17} T^{5} +$$$$39\!\cdots\!16$$$$p^{34} T^{6} - 46481993767813320 p^{51} T^{7} + p^{68} T^{8} )^{2}$$
97 $$1 -$$$$28\!\cdots\!80$$$$T^{2} +$$$$41\!\cdots\!76$$$$T^{4} -$$$$38\!\cdots\!60$$$$T^{6} +$$$$26\!\cdots\!66$$$$T^{8} -$$$$38\!\cdots\!60$$$$p^{34} T^{10} +$$$$41\!\cdots\!76$$$$p^{68} T^{12} -$$$$28\!\cdots\!80$$$$p^{102} T^{14} + p^{136} T^{16}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}