# Properties

 Degree 12 Conductor $5^{6}$ Sign $1$ Motivic weight 33 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.47e5·2-s + 2.65e7·3-s − 1.53e8·4-s + 9.15e11·5-s + 3.90e12·6-s + 1.91e13·7-s − 4.87e14·8-s − 9.52e15·9-s + 1.34e17·10-s + 1.50e17·11-s − 4.06e15·12-s − 1.40e18·13-s + 2.81e18·14-s + 2.42e19·15-s + 2.07e18·16-s + 1.34e20·17-s − 1.40e21·18-s + 1.71e21·19-s − 1.40e20·20-s + 5.06e20·21-s + 2.22e22·22-s + 6.87e21·23-s − 1.29e22·24-s + 4.88e23·25-s − 2.06e23·26-s − 4.21e23·27-s − 2.93e21·28-s + ⋯
 L(s)  = 1 + 1.58·2-s + 0.355·3-s − 0.0178·4-s + 2.68·5-s + 0.565·6-s + 0.217·7-s − 0.612·8-s − 1.71·9-s + 4.26·10-s + 0.989·11-s − 0.00635·12-s − 0.584·13-s + 0.345·14-s + 0.954·15-s + 0.0280·16-s + 0.669·17-s − 2.72·18-s + 1.36·19-s − 0.0479·20-s + 0.0773·21-s + 1.57·22-s + 0.233·23-s − 0.217·24-s + 21/5·25-s − 0.929·26-s − 1.01·27-s − 0.00388·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+33/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$12$$ $$N$$ = $$15625$$    =    $$5^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$33$$ character : induced by $\chi_{5} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(12,\ 15625,\ (\ :[33/2]^{6}),\ 1)$ $L(17)$ $\approx$ $87.51151057$ $L(\frac12)$ $\approx$ $87.51151057$ $L(\frac{35}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 5$,$$F_p(T)$$ is a polynomial of degree 12. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad5 $$( 1 - p^{16} T )^{6}$$
good2 $$1 - 73675 p T + 2733186525 p^{3} T^{2} - 21537675951275 p^{7} T^{3} + 5121518228312647 p^{16} T^{4} - 2120910646147229675 p^{24} T^{5} +$$$$15\!\cdots\!25$$$$p^{31} T^{6} - 2120910646147229675 p^{57} T^{7} + 5121518228312647 p^{82} T^{8} - 21537675951275 p^{106} T^{9} + 2733186525 p^{135} T^{10} - 73675 p^{166} T^{11} + p^{198} T^{12}$$
3 $$1 - 26513900 T + 3409207951532950 p T^{2} -$$$$12\!\cdots\!00$$$$p^{4} T^{3} +$$$$50\!\cdots\!89$$$$p^{9} T^{4} -$$$$60\!\cdots\!00$$$$p^{15} T^{5} +$$$$18\!\cdots\!00$$$$p^{22} T^{6} -$$$$60\!\cdots\!00$$$$p^{48} T^{7} +$$$$50\!\cdots\!89$$$$p^{75} T^{8} -$$$$12\!\cdots\!00$$$$p^{103} T^{9} + 3409207951532950 p^{133} T^{10} - 26513900 p^{165} T^{11} + p^{198} T^{12}$$
7 $$1 - 19113832847500 T +$$$$38\!\cdots\!50$$$$p T^{2} -$$$$20\!\cdots\!00$$$$p^{2} T^{3} +$$$$10\!\cdots\!29$$$$p^{3} T^{4} -$$$$12\!\cdots\!00$$$$p^{6} T^{5} +$$$$12\!\cdots\!00$$$$p^{10} T^{6} -$$$$12\!\cdots\!00$$$$p^{39} T^{7} +$$$$10\!\cdots\!29$$$$p^{69} T^{8} -$$$$20\!\cdots\!00$$$$p^{101} T^{9} +$$$$38\!\cdots\!50$$$$p^{133} T^{10} - 19113832847500 p^{165} T^{11} + p^{198} T^{12}$$
11 $$1 - 113268892979632 p^{3} T +$$$$62\!\cdots\!26$$$$p^{2} T^{2} -$$$$36\!\cdots\!20$$$$p^{3} T^{3} +$$$$13\!\cdots\!45$$$$p^{5} T^{4} -$$$$59\!\cdots\!92$$$$p^{5} T^{5} +$$$$25\!\cdots\!44$$$$p^{6} T^{6} -$$$$59\!\cdots\!92$$$$p^{38} T^{7} +$$$$13\!\cdots\!45$$$$p^{71} T^{8} -$$$$36\!\cdots\!20$$$$p^{102} T^{9} +$$$$62\!\cdots\!26$$$$p^{134} T^{10} - 113268892979632 p^{168} T^{11} + p^{198} T^{12}$$
13 $$1 + 1403095636752804700 T +$$$$22\!\cdots\!50$$$$T^{2} +$$$$17\!\cdots\!00$$$$p T^{3} +$$$$11\!\cdots\!91$$$$p^{3} T^{4} +$$$$57\!\cdots\!00$$$$p^{5} T^{5} +$$$$22\!\cdots\!00$$$$p^{8} T^{6} +$$$$57\!\cdots\!00$$$$p^{38} T^{7} +$$$$11\!\cdots\!91$$$$p^{69} T^{8} +$$$$17\!\cdots\!00$$$$p^{100} T^{9} +$$$$22\!\cdots\!50$$$$p^{132} T^{10} + 1403095636752804700 p^{165} T^{11} + p^{198} T^{12}$$
17 $$1 -$$$$13\!\cdots\!00$$$$T +$$$$73\!\cdots\!50$$$$T^{2} -$$$$17\!\cdots\!00$$$$p T^{3} +$$$$83\!\cdots\!63$$$$p^{2} T^{4} +$$$$51\!\cdots\!00$$$$p^{3} T^{5} +$$$$56\!\cdots\!00$$$$p^{4} T^{6} +$$$$51\!\cdots\!00$$$$p^{36} T^{7} +$$$$83\!\cdots\!63$$$$p^{68} T^{8} -$$$$17\!\cdots\!00$$$$p^{100} T^{9} +$$$$73\!\cdots\!50$$$$p^{132} T^{10} -$$$$13\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
19 $$1 -$$$$17\!\cdots\!00$$$$T +$$$$41\!\cdots\!54$$$$T^{2} -$$$$30\!\cdots\!00$$$$p T^{3} +$$$$33\!\cdots\!15$$$$p^{2} T^{4} -$$$$21\!\cdots\!00$$$$p^{3} T^{5} +$$$$17\!\cdots\!80$$$$p^{4} T^{6} -$$$$21\!\cdots\!00$$$$p^{36} T^{7} +$$$$33\!\cdots\!15$$$$p^{68} T^{8} -$$$$30\!\cdots\!00$$$$p^{100} T^{9} +$$$$41\!\cdots\!54$$$$p^{132} T^{10} -$$$$17\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
23 $$1 -$$$$29\!\cdots\!00$$$$p T +$$$$22\!\cdots\!50$$$$T^{2} -$$$$89\!\cdots\!00$$$$p T^{3} +$$$$69\!\cdots\!23$$$$p^{2} T^{4} -$$$$23\!\cdots\!00$$$$p^{3} T^{5} +$$$$12\!\cdots\!00$$$$p^{4} T^{6} -$$$$23\!\cdots\!00$$$$p^{36} T^{7} +$$$$69\!\cdots\!23$$$$p^{68} T^{8} -$$$$89\!\cdots\!00$$$$p^{100} T^{9} +$$$$22\!\cdots\!50$$$$p^{132} T^{10} -$$$$29\!\cdots\!00$$$$p^{166} T^{11} + p^{198} T^{12}$$
29 $$1 -$$$$13\!\cdots\!00$$$$T +$$$$12\!\cdots\!46$$$$p T^{2} -$$$$73\!\cdots\!00$$$$p^{2} T^{3} +$$$$39\!\cdots\!35$$$$p^{3} T^{4} -$$$$25\!\cdots\!00$$$$p^{4} T^{5} +$$$$10\!\cdots\!20$$$$p^{5} T^{6} -$$$$25\!\cdots\!00$$$$p^{37} T^{7} +$$$$39\!\cdots\!35$$$$p^{69} T^{8} -$$$$73\!\cdots\!00$$$$p^{101} T^{9} +$$$$12\!\cdots\!46$$$$p^{133} T^{10} -$$$$13\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
31 $$1 -$$$$34\!\cdots\!92$$$$p T +$$$$44\!\cdots\!66$$$$p^{3} T^{2} -$$$$30\!\cdots\!20$$$$p^{3} T^{3} +$$$$66\!\cdots\!95$$$$p^{4} T^{4} -$$$$10\!\cdots\!92$$$$p^{5} T^{5} +$$$$15\!\cdots\!84$$$$p^{6} T^{6} -$$$$10\!\cdots\!92$$$$p^{38} T^{7} +$$$$66\!\cdots\!95$$$$p^{70} T^{8} -$$$$30\!\cdots\!20$$$$p^{102} T^{9} +$$$$44\!\cdots\!66$$$$p^{135} T^{10} -$$$$34\!\cdots\!92$$$$p^{166} T^{11} + p^{198} T^{12}$$
37 $$1 -$$$$27\!\cdots\!00$$$$T +$$$$39\!\cdots\!50$$$$T^{2} -$$$$38\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!27$$$$T^{4} -$$$$26\!\cdots\!00$$$$T^{5} +$$$$21\!\cdots\!00$$$$T^{6} -$$$$26\!\cdots\!00$$$$p^{33} T^{7} +$$$$31\!\cdots\!27$$$$p^{66} T^{8} -$$$$38\!\cdots\!00$$$$p^{99} T^{9} +$$$$39\!\cdots\!50$$$$p^{132} T^{10} -$$$$27\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
41 $$1 -$$$$44\!\cdots\!32$$$$T +$$$$40\!\cdots\!86$$$$T^{2} -$$$$20\!\cdots\!20$$$$T^{3} +$$$$95\!\cdots\!95$$$$T^{4} -$$$$34\!\cdots\!92$$$$T^{5} +$$$$18\!\cdots\!64$$$$T^{6} -$$$$34\!\cdots\!92$$$$p^{33} T^{7} +$$$$95\!\cdots\!95$$$$p^{66} T^{8} -$$$$20\!\cdots\!20$$$$p^{99} T^{9} +$$$$40\!\cdots\!86$$$$p^{132} T^{10} -$$$$44\!\cdots\!32$$$$p^{165} T^{11} + p^{198} T^{12}$$
43 $$1 -$$$$22\!\cdots\!00$$$$T +$$$$61\!\cdots\!50$$$$T^{2} -$$$$89\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!47$$$$T^{4} -$$$$14\!\cdots\!00$$$$T^{5} +$$$$15\!\cdots\!00$$$$T^{6} -$$$$14\!\cdots\!00$$$$p^{33} T^{7} +$$$$13\!\cdots\!47$$$$p^{66} T^{8} -$$$$89\!\cdots\!00$$$$p^{99} T^{9} +$$$$61\!\cdots\!50$$$$p^{132} T^{10} -$$$$22\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
47 $$1 -$$$$38\!\cdots\!00$$$$T +$$$$24\!\cdots\!50$$$$T^{2} -$$$$21\!\cdots\!00$$$$T^{3} -$$$$65\!\cdots\!13$$$$T^{4} +$$$$64\!\cdots\!00$$$$T^{5} -$$$$42\!\cdots\!00$$$$T^{6} +$$$$64\!\cdots\!00$$$$p^{33} T^{7} -$$$$65\!\cdots\!13$$$$p^{66} T^{8} -$$$$21\!\cdots\!00$$$$p^{99} T^{9} +$$$$24\!\cdots\!50$$$$p^{132} T^{10} -$$$$38\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
53 $$1 -$$$$47\!\cdots\!00$$$$T +$$$$23\!\cdots\!50$$$$T^{2} -$$$$65\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!87$$$$T^{4} -$$$$83\!\cdots\!00$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} -$$$$83\!\cdots\!00$$$$p^{33} T^{7} +$$$$25\!\cdots\!87$$$$p^{66} T^{8} -$$$$65\!\cdots\!00$$$$p^{99} T^{9} +$$$$23\!\cdots\!50$$$$p^{132} T^{10} -$$$$47\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
59 $$1 +$$$$17\!\cdots\!00$$$$T +$$$$85\!\cdots\!74$$$$T^{2} +$$$$90\!\cdots\!00$$$$T^{3} +$$$$19\!\cdots\!15$$$$T^{4} +$$$$91\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!80$$$$T^{6} +$$$$91\!\cdots\!00$$$$p^{33} T^{7} +$$$$19\!\cdots\!15$$$$p^{66} T^{8} +$$$$90\!\cdots\!00$$$$p^{99} T^{9} +$$$$85\!\cdots\!74$$$$p^{132} T^{10} +$$$$17\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
61 $$1 +$$$$28\!\cdots\!08$$$$T +$$$$42\!\cdots\!46$$$$T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$80\!\cdots\!95$$$$T^{4} +$$$$16\!\cdots\!08$$$$T^{5} +$$$$85\!\cdots\!84$$$$T^{6} +$$$$16\!\cdots\!08$$$$p^{33} T^{7} +$$$$80\!\cdots\!95$$$$p^{66} T^{8} +$$$$10\!\cdots\!80$$$$p^{99} T^{9} +$$$$42\!\cdots\!46$$$$p^{132} T^{10} +$$$$28\!\cdots\!08$$$$p^{165} T^{11} + p^{198} T^{12}$$
67 $$1 -$$$$33\!\cdots\!00$$$$T +$$$$90\!\cdots\!50$$$$T^{2} -$$$$14\!\cdots\!00$$$$T^{3} +$$$$22\!\cdots\!07$$$$T^{4} -$$$$24\!\cdots\!00$$$$T^{5} +$$$$34\!\cdots\!00$$$$T^{6} -$$$$24\!\cdots\!00$$$$p^{33} T^{7} +$$$$22\!\cdots\!07$$$$p^{66} T^{8} -$$$$14\!\cdots\!00$$$$p^{99} T^{9} +$$$$90\!\cdots\!50$$$$p^{132} T^{10} -$$$$33\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
71 $$1 -$$$$84\!\cdots\!72$$$$T +$$$$11\!\cdots\!06$$$$p T^{2} -$$$$44\!\cdots\!20$$$$T^{3} +$$$$24\!\cdots\!95$$$$T^{4} -$$$$99\!\cdots\!92$$$$T^{5} +$$$$39\!\cdots\!44$$$$T^{6} -$$$$99\!\cdots\!92$$$$p^{33} T^{7} +$$$$24\!\cdots\!95$$$$p^{66} T^{8} -$$$$44\!\cdots\!20$$$$p^{99} T^{9} +$$$$11\!\cdots\!06$$$$p^{133} T^{10} -$$$$84\!\cdots\!72$$$$p^{165} T^{11} + p^{198} T^{12}$$
73 $$1 +$$$$17\!\cdots\!00$$$$T +$$$$12\!\cdots\!50$$$$T^{2} +$$$$51\!\cdots\!00$$$$T^{3} +$$$$73\!\cdots\!67$$$$T^{4} +$$$$15\!\cdots\!00$$$$T^{5} +$$$$28\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!00$$$$p^{33} T^{7} +$$$$73\!\cdots\!67$$$$p^{66} T^{8} +$$$$51\!\cdots\!00$$$$p^{99} T^{9} +$$$$12\!\cdots\!50$$$$p^{132} T^{10} +$$$$17\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
79 $$1 +$$$$22\!\cdots\!00$$$$T +$$$$23\!\cdots\!34$$$$T^{2} +$$$$43\!\cdots\!00$$$$T^{3} +$$$$23\!\cdots\!15$$$$T^{4} +$$$$34\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!80$$$$T^{6} +$$$$34\!\cdots\!00$$$$p^{33} T^{7} +$$$$23\!\cdots\!15$$$$p^{66} T^{8} +$$$$43\!\cdots\!00$$$$p^{99} T^{9} +$$$$23\!\cdots\!34$$$$p^{132} T^{10} +$$$$22\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
83 $$1 +$$$$47\!\cdots\!00$$$$T +$$$$88\!\cdots\!50$$$$T^{2} +$$$$37\!\cdots\!00$$$$T^{3} +$$$$39\!\cdots\!07$$$$T^{4} +$$$$14\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!00$$$$p^{33} T^{7} +$$$$39\!\cdots\!07$$$$p^{66} T^{8} +$$$$37\!\cdots\!00$$$$p^{99} T^{9} +$$$$88\!\cdots\!50$$$$p^{132} T^{10} +$$$$47\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
89 $$1 +$$$$33\!\cdots\!00$$$$T +$$$$10\!\cdots\!14$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!15$$$$T^{4} +$$$$68\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!80$$$$T^{6} +$$$$68\!\cdots\!00$$$$p^{33} T^{7} +$$$$40\!\cdots\!15$$$$p^{66} T^{8} +$$$$20\!\cdots\!00$$$$p^{99} T^{9} +$$$$10\!\cdots\!14$$$$p^{132} T^{10} +$$$$33\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
97 $$1 -$$$$25\!\cdots\!00$$$$T +$$$$42\!\cdots\!50$$$$T^{2} -$$$$52\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!87$$$$T^{4} -$$$$39\!\cdots\!00$$$$T^{5} +$$$$26\!\cdots\!00$$$$T^{6} -$$$$39\!\cdots\!00$$$$p^{33} T^{7} +$$$$50\!\cdots\!87$$$$p^{66} T^{8} -$$$$52\!\cdots\!00$$$$p^{99} T^{9} +$$$$42\!\cdots\!50$$$$p^{132} T^{10} -$$$$25\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}