# Properties

 Degree $24$ Conductor $5.960\times 10^{16}$ Sign $1$ Motivic weight $33$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2.20e10·4-s + 1.97e16·9-s + 3.01e17·11-s + 1.47e20·16-s − 3.42e21·19-s − 2.61e24·29-s + 2.16e25·31-s + 4.34e26·36-s + 8.96e26·41-s + 6.63e27·44-s + 5.35e28·49-s + 3.56e29·59-s − 5.64e29·61-s − 7.63e29·64-s + 1.69e31·71-s − 7.53e31·76-s + 4.41e31·79-s + 9.06e31·81-s + 6.72e32·89-s + 5.95e33·99-s + 1.96e33·101-s + 1.33e34·109-s − 5.74e34·116-s − 8.33e34·121-s + 4.76e35·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 + 2.56·4-s + 3.55·9-s + 1.97·11-s + 2.00·16-s − 2.72·19-s − 1.93·29-s + 5.34·31-s + 9.10·36-s + 2.19·41-s + 5.07·44-s + 6.92·49-s + 2.15·59-s − 1.96·61-s − 1.20·64-s + 4.81·71-s − 6.97·76-s + 2.15·79-s + 2.93·81-s + 4.60·89-s + 7.02·99-s + 1.66·101-s + 3.20·109-s − 4.96·116-s − 3.58·121-s + 13.7·124-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(34-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24}\right)^{s/2} \, \Gamma_{\C}(s+33/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$5^{24}$$ Sign: $1$ Motivic weight: $$33$$ Character: induced by $\chi_{25} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 5^{24} ,\ ( \ : [33/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(17)$$ $$\approx$$ $$163.0489029$$ $$L(\frac12)$$ $$\approx$$ $$163.0489029$$ $$L(\frac{35}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
good2 $$1 - 5504740475 p^{2} T^{2} + 5264871706344391681 p^{6} T^{4} -$$$$20\!\cdots\!75$$$$p^{14} T^{6} +$$$$90\!\cdots\!35$$$$p^{38} T^{8} -$$$$24\!\cdots\!25$$$$p^{49} T^{10} +$$$$17\!\cdots\!45$$$$p^{62} T^{12} -$$$$24\!\cdots\!25$$$$p^{115} T^{14} +$$$$90\!\cdots\!35$$$$p^{170} T^{16} -$$$$20\!\cdots\!75$$$$p^{212} T^{18} + 5264871706344391681 p^{270} T^{20} - 5504740475 p^{332} T^{22} + p^{396} T^{24}$$
3 $$1 - 19752260815987700 T^{2} +$$$$33\!\cdots\!86$$$$p^{2} T^{4} -$$$$47\!\cdots\!00$$$$p^{8} T^{6} +$$$$70\!\cdots\!35$$$$p^{18} T^{8} -$$$$94\!\cdots\!00$$$$p^{30} T^{10} +$$$$12\!\cdots\!80$$$$p^{44} T^{12} -$$$$94\!\cdots\!00$$$$p^{96} T^{14} +$$$$70\!\cdots\!35$$$$p^{150} T^{16} -$$$$47\!\cdots\!00$$$$p^{206} T^{18} +$$$$33\!\cdots\!86$$$$p^{266} T^{20} - 19752260815987700 p^{330} T^{22} + p^{396} T^{24}$$
7 $$1 -$$$$53\!\cdots\!00$$$$T^{2} +$$$$29\!\cdots\!06$$$$p^{2} T^{4} -$$$$10\!\cdots\!00$$$$p^{4} T^{6} +$$$$28\!\cdots\!35$$$$p^{6} T^{8} -$$$$25\!\cdots\!00$$$$p^{12} T^{10} +$$$$37\!\cdots\!80$$$$p^{20} T^{12} -$$$$25\!\cdots\!00$$$$p^{78} T^{14} +$$$$28\!\cdots\!35$$$$p^{138} T^{16} -$$$$10\!\cdots\!00$$$$p^{202} T^{18} +$$$$29\!\cdots\!06$$$$p^{266} T^{20} -$$$$53\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
11 $$( 1 - 113268892979632 p^{3} T +$$$$62\!\cdots\!26$$$$p^{2} T^{2} -$$$$36\!\cdots\!20$$$$p^{3} T^{3} +$$$$13\!\cdots\!45$$$$p^{5} T^{4} -$$$$59\!\cdots\!92$$$$p^{5} T^{5} +$$$$25\!\cdots\!44$$$$p^{6} T^{6} -$$$$59\!\cdots\!92$$$$p^{38} T^{7} +$$$$13\!\cdots\!45$$$$p^{71} T^{8} -$$$$36\!\cdots\!20$$$$p^{102} T^{9} +$$$$62\!\cdots\!26$$$$p^{134} T^{10} - 113268892979632 p^{168} T^{11} + p^{198} T^{12} )^{2}$$
13 $$1 -$$$$43\!\cdots\!00$$$$T^{2} +$$$$95\!\cdots\!54$$$$T^{4} -$$$$82\!\cdots\!00$$$$p^{2} T^{6} +$$$$31\!\cdots\!35$$$$p^{6} T^{8} -$$$$89\!\cdots\!00$$$$p^{10} T^{10} +$$$$12\!\cdots\!80$$$$p^{16} T^{12} -$$$$89\!\cdots\!00$$$$p^{76} T^{14} +$$$$31\!\cdots\!35$$$$p^{138} T^{16} -$$$$82\!\cdots\!00$$$$p^{200} T^{18} +$$$$95\!\cdots\!54$$$$p^{264} T^{20} -$$$$43\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
17 $$1 -$$$$12\!\cdots\!00$$$$T^{2} +$$$$94\!\cdots\!14$$$$T^{4} -$$$$17\!\cdots\!00$$$$p^{2} T^{6} +$$$$29\!\cdots\!15$$$$p^{4} T^{8} -$$$$47\!\cdots\!00$$$$p^{6} T^{10} +$$$$68\!\cdots\!80$$$$p^{8} T^{12} -$$$$47\!\cdots\!00$$$$p^{72} T^{14} +$$$$29\!\cdots\!15$$$$p^{136} T^{16} -$$$$17\!\cdots\!00$$$$p^{200} T^{18} +$$$$94\!\cdots\!14$$$$p^{264} T^{20} -$$$$12\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
19 $$( 1 +$$$$17\!\cdots\!00$$$$T +$$$$41\!\cdots\!54$$$$T^{2} +$$$$30\!\cdots\!00$$$$p T^{3} +$$$$33\!\cdots\!15$$$$p^{2} T^{4} +$$$$21\!\cdots\!00$$$$p^{3} T^{5} +$$$$17\!\cdots\!80$$$$p^{4} T^{6} +$$$$21\!\cdots\!00$$$$p^{36} T^{7} +$$$$33\!\cdots\!15$$$$p^{68} T^{8} +$$$$30\!\cdots\!00$$$$p^{100} T^{9} +$$$$41\!\cdots\!54$$$$p^{132} T^{10} +$$$$17\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
23 $$1 -$$$$44\!\cdots\!00$$$$T^{2} +$$$$12\!\cdots\!34$$$$T^{4} -$$$$43\!\cdots\!00$$$$p^{2} T^{6} +$$$$11\!\cdots\!15$$$$p^{4} T^{8} -$$$$26\!\cdots\!00$$$$p^{6} T^{10} +$$$$48\!\cdots\!80$$$$p^{8} T^{12} -$$$$26\!\cdots\!00$$$$p^{72} T^{14} +$$$$11\!\cdots\!15$$$$p^{136} T^{16} -$$$$43\!\cdots\!00$$$$p^{200} T^{18} +$$$$12\!\cdots\!34$$$$p^{264} T^{20} -$$$$44\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
29 $$( 1 +$$$$13\!\cdots\!00$$$$T +$$$$12\!\cdots\!46$$$$p T^{2} +$$$$73\!\cdots\!00$$$$p^{2} T^{3} +$$$$39\!\cdots\!35$$$$p^{3} T^{4} +$$$$25\!\cdots\!00$$$$p^{4} T^{5} +$$$$10\!\cdots\!20$$$$p^{5} T^{6} +$$$$25\!\cdots\!00$$$$p^{37} T^{7} +$$$$39\!\cdots\!35$$$$p^{69} T^{8} +$$$$73\!\cdots\!00$$$$p^{101} T^{9} +$$$$12\!\cdots\!46$$$$p^{133} T^{10} +$$$$13\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
31 $$( 1 -$$$$34\!\cdots\!92$$$$p T +$$$$44\!\cdots\!66$$$$p^{3} T^{2} -$$$$30\!\cdots\!20$$$$p^{3} T^{3} +$$$$66\!\cdots\!95$$$$p^{4} T^{4} -$$$$10\!\cdots\!92$$$$p^{5} T^{5} +$$$$15\!\cdots\!84$$$$p^{6} T^{6} -$$$$10\!\cdots\!92$$$$p^{38} T^{7} +$$$$66\!\cdots\!95$$$$p^{70} T^{8} -$$$$30\!\cdots\!20$$$$p^{102} T^{9} +$$$$44\!\cdots\!66$$$$p^{135} T^{10} -$$$$34\!\cdots\!92$$$$p^{166} T^{11} + p^{198} T^{12} )^{2}$$
37 $$1 -$$$$33\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!54$$$$T^{4} -$$$$43\!\cdots\!00$$$$T^{6} +$$$$66\!\cdots\!15$$$$T^{8} -$$$$23\!\cdots\!00$$$$T^{10} +$$$$25\!\cdots\!80$$$$T^{12} -$$$$23\!\cdots\!00$$$$p^{66} T^{14} +$$$$66\!\cdots\!15$$$$p^{132} T^{16} -$$$$43\!\cdots\!00$$$$p^{198} T^{18} +$$$$11\!\cdots\!54$$$$p^{264} T^{20} -$$$$33\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
41 $$( 1 -$$$$44\!\cdots\!32$$$$T +$$$$40\!\cdots\!86$$$$T^{2} -$$$$20\!\cdots\!20$$$$T^{3} +$$$$95\!\cdots\!95$$$$T^{4} -$$$$34\!\cdots\!92$$$$T^{5} +$$$$18\!\cdots\!64$$$$T^{6} -$$$$34\!\cdots\!92$$$$p^{33} T^{7} +$$$$95\!\cdots\!95$$$$p^{66} T^{8} -$$$$20\!\cdots\!20$$$$p^{99} T^{9} +$$$$40\!\cdots\!86$$$$p^{132} T^{10} -$$$$44\!\cdots\!32$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
43 $$1 -$$$$71\!\cdots\!00$$$$T^{2} +$$$$24\!\cdots\!94$$$$T^{4} -$$$$52\!\cdots\!00$$$$T^{6} +$$$$81\!\cdots\!15$$$$T^{8} -$$$$95\!\cdots\!00$$$$T^{10} +$$$$86\!\cdots\!80$$$$T^{12} -$$$$95\!\cdots\!00$$$$p^{66} T^{14} +$$$$81\!\cdots\!15$$$$p^{132} T^{16} -$$$$52\!\cdots\!00$$$$p^{198} T^{18} +$$$$24\!\cdots\!94$$$$p^{264} T^{20} -$$$$71\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
47 $$1 -$$$$34\!\cdots\!00$$$$T^{2} +$$$$31\!\cdots\!74$$$$T^{4} +$$$$71\!\cdots\!00$$$$T^{6} -$$$$13\!\cdots\!85$$$$T^{8} -$$$$18\!\cdots\!00$$$$T^{10} +$$$$68\!\cdots\!80$$$$T^{12} -$$$$18\!\cdots\!00$$$$p^{66} T^{14} -$$$$13\!\cdots\!85$$$$p^{132} T^{16} +$$$$71\!\cdots\!00$$$$p^{198} T^{18} +$$$$31\!\cdots\!74$$$$p^{264} T^{20} -$$$$34\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
53 $$1 -$$$$24\!\cdots\!00$$$$T^{2} +$$$$41\!\cdots\!74$$$$T^{4} -$$$$49\!\cdots\!00$$$$T^{6} +$$$$49\!\cdots\!15$$$$T^{8} -$$$$41\!\cdots\!00$$$$T^{10} +$$$$34\!\cdots\!80$$$$T^{12} -$$$$41\!\cdots\!00$$$$p^{66} T^{14} +$$$$49\!\cdots\!15$$$$p^{132} T^{16} -$$$$49\!\cdots\!00$$$$p^{198} T^{18} +$$$$41\!\cdots\!74$$$$p^{264} T^{20} -$$$$24\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
59 $$( 1 -$$$$17\!\cdots\!00$$$$T +$$$$85\!\cdots\!74$$$$T^{2} -$$$$90\!\cdots\!00$$$$T^{3} +$$$$19\!\cdots\!15$$$$T^{4} -$$$$91\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!80$$$$T^{6} -$$$$91\!\cdots\!00$$$$p^{33} T^{7} +$$$$19\!\cdots\!15$$$$p^{66} T^{8} -$$$$90\!\cdots\!00$$$$p^{99} T^{9} +$$$$85\!\cdots\!74$$$$p^{132} T^{10} -$$$$17\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
61 $$( 1 +$$$$28\!\cdots\!08$$$$T +$$$$42\!\cdots\!46$$$$T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$80\!\cdots\!95$$$$T^{4} +$$$$16\!\cdots\!08$$$$T^{5} +$$$$85\!\cdots\!84$$$$T^{6} +$$$$16\!\cdots\!08$$$$p^{33} T^{7} +$$$$80\!\cdots\!95$$$$p^{66} T^{8} +$$$$10\!\cdots\!80$$$$p^{99} T^{9} +$$$$42\!\cdots\!46$$$$p^{132} T^{10} +$$$$28\!\cdots\!08$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
67 $$1 -$$$$68\!\cdots\!00$$$$T^{2} +$$$$26\!\cdots\!14$$$$T^{4} -$$$$83\!\cdots\!00$$$$T^{6} +$$$$22\!\cdots\!15$$$$T^{8} -$$$$51\!\cdots\!00$$$$T^{10} +$$$$98\!\cdots\!80$$$$T^{12} -$$$$51\!\cdots\!00$$$$p^{66} T^{14} +$$$$22\!\cdots\!15$$$$p^{132} T^{16} -$$$$83\!\cdots\!00$$$$p^{198} T^{18} +$$$$26\!\cdots\!14$$$$p^{264} T^{20} -$$$$68\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
71 $$( 1 -$$$$84\!\cdots\!72$$$$T +$$$$11\!\cdots\!06$$$$p T^{2} -$$$$44\!\cdots\!20$$$$T^{3} +$$$$24\!\cdots\!95$$$$T^{4} -$$$$99\!\cdots\!92$$$$T^{5} +$$$$39\!\cdots\!44$$$$T^{6} -$$$$99\!\cdots\!92$$$$p^{33} T^{7} +$$$$24\!\cdots\!95$$$$p^{66} T^{8} -$$$$44\!\cdots\!20$$$$p^{99} T^{9} +$$$$11\!\cdots\!06$$$$p^{133} T^{10} -$$$$84\!\cdots\!72$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
73 $$1 -$$$$24\!\cdots\!00$$$$T^{2} +$$$$29\!\cdots\!34$$$$T^{4} -$$$$23\!\cdots\!00$$$$T^{6} +$$$$13\!\cdots\!15$$$$T^{8} -$$$$61\!\cdots\!00$$$$T^{10} +$$$$21\!\cdots\!80$$$$T^{12} -$$$$61\!\cdots\!00$$$$p^{66} T^{14} +$$$$13\!\cdots\!15$$$$p^{132} T^{16} -$$$$23\!\cdots\!00$$$$p^{198} T^{18} +$$$$29\!\cdots\!34$$$$p^{264} T^{20} -$$$$24\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
79 $$( 1 -$$$$22\!\cdots\!00$$$$T +$$$$23\!\cdots\!34$$$$T^{2} -$$$$43\!\cdots\!00$$$$T^{3} +$$$$23\!\cdots\!15$$$$T^{4} -$$$$34\!\cdots\!00$$$$T^{5} +$$$$13\!\cdots\!80$$$$T^{6} -$$$$34\!\cdots\!00$$$$p^{33} T^{7} +$$$$23\!\cdots\!15$$$$p^{66} T^{8} -$$$$43\!\cdots\!00$$$$p^{99} T^{9} +$$$$23\!\cdots\!34$$$$p^{132} T^{10} -$$$$22\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
83 $$1 -$$$$15\!\cdots\!00$$$$T^{2} +$$$$12\!\cdots\!14$$$$T^{4} -$$$$62\!\cdots\!00$$$$T^{6} +$$$$24\!\cdots\!15$$$$T^{8} -$$$$71\!\cdots\!00$$$$T^{10} +$$$$17\!\cdots\!80$$$$T^{12} -$$$$71\!\cdots\!00$$$$p^{66} T^{14} +$$$$24\!\cdots\!15$$$$p^{132} T^{16} -$$$$62\!\cdots\!00$$$$p^{198} T^{18} +$$$$12\!\cdots\!14$$$$p^{264} T^{20} -$$$$15\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
89 $$( 1 -$$$$33\!\cdots\!00$$$$T +$$$$10\!\cdots\!14$$$$T^{2} -$$$$20\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!15$$$$T^{4} -$$$$68\!\cdots\!00$$$$T^{5} +$$$$10\!\cdots\!80$$$$T^{6} -$$$$68\!\cdots\!00$$$$p^{33} T^{7} +$$$$40\!\cdots\!15$$$$p^{66} T^{8} -$$$$20\!\cdots\!00$$$$p^{99} T^{9} +$$$$10\!\cdots\!14$$$$p^{132} T^{10} -$$$$33\!\cdots\!00$$$$p^{165} T^{11} + p^{198} T^{12} )^{2}$$
97 $$1 -$$$$19\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!74$$$$T^{4} -$$$$86\!\cdots\!00$$$$T^{6} +$$$$38\!\cdots\!15$$$$T^{8} -$$$$17\!\cdots\!00$$$$T^{10} +$$$$73\!\cdots\!80$$$$T^{12} -$$$$17\!\cdots\!00$$$$p^{66} T^{14} +$$$$38\!\cdots\!15$$$$p^{132} T^{16} -$$$$86\!\cdots\!00$$$$p^{198} T^{18} +$$$$16\!\cdots\!74$$$$p^{264} T^{20} -$$$$19\!\cdots\!00$$$$p^{330} T^{22} + p^{396} T^{24}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$