# Properties

 Label 40-384e20-1.1-c9e20-0-0 Degree $40$ Conductor $4.860\times 10^{51}$ Sign $1$ Analytic cond. $8.38249\times 10^{45}$ Root an. cond. $14.0632$ Motivic weight $9$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6.56e4·9-s − 9.05e5·17-s + 1.46e7·25-s + 7.42e7·41-s − 2.77e8·49-s − 8.95e8·73-s + 2.36e9·81-s + 8.82e8·89-s + 4.33e8·97-s + 3.53e9·113-s + 1.11e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 5.94e10·153-s + 157-s + 163-s + 167-s + 6.58e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 3.33·9-s − 2.63·17-s + 7.52·25-s + 4.10·41-s − 6.86·49-s − 3.68·73-s + 55/9·81-s + 1.49·89-s + 0.497·97-s + 2.03·113-s + 4.74·121-s + 8.76·153-s + 6.21·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$40$$ Conductor: $$2^{140} \cdot 3^{20}$$ Sign: $1$ Analytic conductor: $$8.38249\times 10^{45}$$ Root analytic conductor: $$14.0632$$ Motivic weight: $$9$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [9/2]^{20} ),\ 1 )$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$20.62785461$$ $$L(\frac12)$$ $$\approx$$ $$20.62785461$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$( 1 + p^{8} T^{2} )^{10}$$
good5 $$( 1 - 1468994 p T^{2} + 30651186670437 T^{4} - 96038225164508667192 T^{6} +$$$$19\!\cdots\!62$$$$p^{3} T^{8} -$$$$80\!\cdots\!12$$$$p^{4} T^{10} +$$$$19\!\cdots\!62$$$$p^{21} T^{12} - 96038225164508667192 p^{36} T^{14} + 30651186670437 p^{54} T^{16} - 1468994 p^{73} T^{18} + p^{90} T^{20} )^{2}$$
7 $$( 1 + 138574110 T^{2} + 22746130578923 p^{3} T^{4} +$$$$13\!\cdots\!96$$$$p^{4} T^{6} +$$$$16\!\cdots\!62$$$$p^{6} T^{8} +$$$$16\!\cdots\!72$$$$p^{8} T^{10} +$$$$16\!\cdots\!62$$$$p^{24} T^{12} +$$$$13\!\cdots\!96$$$$p^{40} T^{14} + 22746130578923 p^{57} T^{16} + 138574110 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
11 $$( 1 - 5594405214 T^{2} + 1320109756127459717 T^{4} +$$$$36\!\cdots\!04$$$$T^{6} +$$$$22\!\cdots\!46$$$$T^{8} -$$$$33\!\cdots\!28$$$$T^{10} +$$$$22\!\cdots\!46$$$$p^{18} T^{12} +$$$$36\!\cdots\!04$$$$p^{36} T^{14} + 1320109756127459717 p^{54} T^{16} - 5594405214 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
13 $$( 1 - 32949251106 T^{2} +$$$$66\!\cdots\!85$$$$T^{4} -$$$$11\!\cdots\!80$$$$T^{6} +$$$$15\!\cdots\!98$$$$T^{8} -$$$$17\!\cdots\!84$$$$T^{10} +$$$$15\!\cdots\!98$$$$p^{18} T^{12} -$$$$11\!\cdots\!80$$$$p^{36} T^{14} +$$$$66\!\cdots\!85$$$$p^{54} T^{16} - 32949251106 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
17 $$( 1 + 226442 T + 298338751229 T^{2} + 55181224456447224 T^{3} +$$$$50\!\cdots\!62$$$$T^{4} +$$$$90\!\cdots\!36$$$$T^{5} +$$$$50\!\cdots\!62$$$$p^{9} T^{6} + 55181224456447224 p^{18} T^{7} + 298338751229 p^{27} T^{8} + 226442 p^{36} T^{9} + p^{45} T^{10} )^{4}$$
19 $$( 1 - 2023428505646 T^{2} +$$$$20\!\cdots\!93$$$$T^{4} -$$$$13\!\cdots\!32$$$$T^{6} +$$$$66\!\cdots\!02$$$$T^{8} -$$$$24\!\cdots\!56$$$$T^{10} +$$$$66\!\cdots\!02$$$$p^{18} T^{12} -$$$$13\!\cdots\!32$$$$p^{36} T^{14} +$$$$20\!\cdots\!93$$$$p^{54} T^{16} - 2023428505646 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
23 $$( 1 + 6696077631430 T^{2} +$$$$23\!\cdots\!61$$$$T^{4} +$$$$58\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!74$$$$T^{8} +$$$$25\!\cdots\!16$$$$T^{10} +$$$$12\!\cdots\!74$$$$p^{18} T^{12} +$$$$58\!\cdots\!60$$$$p^{36} T^{14} +$$$$23\!\cdots\!61$$$$p^{54} T^{16} + 6696077631430 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
29 $$( 1 - 111359043735994 T^{2} +$$$$59\!\cdots\!89$$$$T^{4} -$$$$19\!\cdots\!36$$$$T^{6} +$$$$46\!\cdots\!82$$$$T^{8} -$$$$78\!\cdots\!44$$$$T^{10} +$$$$46\!\cdots\!82$$$$p^{18} T^{12} -$$$$19\!\cdots\!36$$$$p^{36} T^{14} +$$$$59\!\cdots\!89$$$$p^{54} T^{16} - 111359043735994 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
31 $$( 1 + 89941603007566 T^{2} +$$$$44\!\cdots\!89$$$$T^{4} +$$$$14\!\cdots\!08$$$$T^{6} +$$$$33\!\cdots\!82$$$$T^{8} +$$$$79\!\cdots\!64$$$$T^{10} +$$$$33\!\cdots\!82$$$$p^{18} T^{12} +$$$$14\!\cdots\!08$$$$p^{36} T^{14} +$$$$44\!\cdots\!89$$$$p^{54} T^{16} + 89941603007566 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
37 $$( 1 - 950204653074226 T^{2} +$$$$43\!\cdots\!69$$$$T^{4} -$$$$12\!\cdots\!40$$$$T^{6} +$$$$26\!\cdots\!14$$$$T^{8} -$$$$40\!\cdots\!72$$$$T^{10} +$$$$26\!\cdots\!14$$$$p^{18} T^{12} -$$$$12\!\cdots\!40$$$$p^{36} T^{14} +$$$$43\!\cdots\!69$$$$p^{54} T^{16} - 950204653074226 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
41 $$( 1 - 18566002 T + 707323868353013 T^{2} -$$$$99\!\cdots\!60$$$$T^{3} +$$$$29\!\cdots\!34$$$$T^{4} -$$$$28\!\cdots\!36$$$$T^{5} +$$$$29\!\cdots\!34$$$$p^{9} T^{6} -$$$$99\!\cdots\!60$$$$p^{18} T^{7} + 707323868353013 p^{27} T^{8} - 18566002 p^{36} T^{9} + p^{45} T^{10} )^{4}$$
43 $$( 1 - 1602548191720990 T^{2} +$$$$12\!\cdots\!29$$$$T^{4} -$$$$67\!\cdots\!48$$$$T^{6} +$$$$39\!\cdots\!02$$$$T^{8} -$$$$21\!\cdots\!44$$$$T^{10} +$$$$39\!\cdots\!02$$$$p^{18} T^{12} -$$$$67\!\cdots\!48$$$$p^{36} T^{14} +$$$$12\!\cdots\!29$$$$p^{54} T^{16} - 1602548191720990 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
47 $$( 1 + 4650398863290934 T^{2} +$$$$10\!\cdots\!29$$$$T^{4} +$$$$17\!\cdots\!80$$$$T^{6} +$$$$24\!\cdots\!02$$$$T^{8} +$$$$29\!\cdots\!96$$$$T^{10} +$$$$24\!\cdots\!02$$$$p^{18} T^{12} +$$$$17\!\cdots\!80$$$$p^{36} T^{14} +$$$$10\!\cdots\!29$$$$p^{54} T^{16} + 4650398863290934 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
53 $$( 1 - 15550103826912298 T^{2} +$$$$11\!\cdots\!21$$$$T^{4} -$$$$63\!\cdots\!88$$$$T^{6} +$$$$28\!\cdots\!74$$$$T^{8} -$$$$10\!\cdots\!96$$$$T^{10} +$$$$28\!\cdots\!74$$$$p^{18} T^{12} -$$$$63\!\cdots\!88$$$$p^{36} T^{14} +$$$$11\!\cdots\!21$$$$p^{54} T^{16} - 15550103826912298 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
59 $$( 1 - 66972384180888126 T^{2} +$$$$21\!\cdots\!29$$$$T^{4} -$$$$43\!\cdots\!24$$$$T^{6} +$$$$60\!\cdots\!46$$$$T^{8} -$$$$60\!\cdots\!72$$$$T^{10} +$$$$60\!\cdots\!46$$$$p^{18} T^{12} -$$$$43\!\cdots\!24$$$$p^{36} T^{14} +$$$$21\!\cdots\!29$$$$p^{54} T^{16} - 66972384180888126 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
61 $$( 1 - 78984113736497442 T^{2} +$$$$28\!\cdots\!73$$$$T^{4} -$$$$64\!\cdots\!84$$$$T^{6} +$$$$10\!\cdots\!90$$$$T^{8} -$$$$13\!\cdots\!16$$$$T^{10} +$$$$10\!\cdots\!90$$$$p^{18} T^{12} -$$$$64\!\cdots\!84$$$$p^{36} T^{14} +$$$$28\!\cdots\!73$$$$p^{54} T^{16} - 78984113736497442 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
67 $$( 1 - 107853403180866830 T^{2} +$$$$66\!\cdots\!65$$$$T^{4} -$$$$31\!\cdots\!60$$$$T^{6} +$$$$11\!\cdots\!50$$$$T^{8} -$$$$34\!\cdots\!28$$$$T^{10} +$$$$11\!\cdots\!50$$$$p^{18} T^{12} -$$$$31\!\cdots\!60$$$$p^{36} T^{14} +$$$$66\!\cdots\!65$$$$p^{54} T^{16} - 107853403180866830 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
71 $$( 1 + 290646683595608230 T^{2} +$$$$41\!\cdots\!05$$$$T^{4} +$$$$39\!\cdots\!20$$$$T^{6} +$$$$26\!\cdots\!10$$$$T^{8} +$$$$14\!\cdots\!80$$$$T^{10} +$$$$26\!\cdots\!10$$$$p^{18} T^{12} +$$$$39\!\cdots\!20$$$$p^{36} T^{14} +$$$$41\!\cdots\!05$$$$p^{54} T^{16} + 290646683595608230 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
73 $$( 1 + 223798474 T + 88967880198633429 T^{2} +$$$$14\!\cdots\!12$$$$T^{3} +$$$$46\!\cdots\!82$$$$T^{4} +$$$$61\!\cdots\!96$$$$T^{5} +$$$$46\!\cdots\!82$$$$p^{9} T^{6} +$$$$14\!\cdots\!12$$$$p^{18} T^{7} + 88967880198633429 p^{27} T^{8} + 223798474 p^{36} T^{9} + p^{45} T^{10} )^{4}$$
79 $$( 1 + 499851356938410286 T^{2} +$$$$10\!\cdots\!73$$$$T^{4} +$$$$11\!\cdots\!76$$$$T^{6} +$$$$43\!\cdots\!38$$$$T^{8} -$$$$16\!\cdots\!40$$$$T^{10} +$$$$43\!\cdots\!38$$$$p^{18} T^{12} +$$$$11\!\cdots\!76$$$$p^{36} T^{14} +$$$$10\!\cdots\!73$$$$p^{54} T^{16} + 499851356938410286 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
83 $$( 1 - 1071966143100584622 T^{2} +$$$$61\!\cdots\!73$$$$T^{4} -$$$$23\!\cdots\!28$$$$T^{6} +$$$$65\!\cdots\!06$$$$T^{8} -$$$$13\!\cdots\!40$$$$T^{10} +$$$$65\!\cdots\!06$$$$p^{18} T^{12} -$$$$23\!\cdots\!28$$$$p^{36} T^{14} +$$$$61\!\cdots\!73$$$$p^{54} T^{16} - 1071966143100584622 p^{72} T^{18} + p^{90} T^{20} )^{2}$$
89 $$( 1 - 220605534 T + 1090737971548238821 T^{2} -$$$$39\!\cdots\!44$$$$T^{3} +$$$$59\!\cdots\!30$$$$T^{4} -$$$$20\!\cdots\!00$$$$T^{5} +$$$$59\!\cdots\!30$$$$p^{9} T^{6} -$$$$39\!\cdots\!44$$$$p^{18} T^{7} + 1090737971548238821 p^{27} T^{8} - 220605534 p^{36} T^{9} + p^{45} T^{10} )^{4}$$
97 $$( 1 - 108420934 T + 2168537688075866253 T^{2} -$$$$48\!\cdots\!24$$$$T^{3} +$$$$25\!\cdots\!62$$$$p T^{4} -$$$$62\!\cdots\!68$$$$T^{5} +$$$$25\!\cdots\!62$$$$p^{10} T^{6} -$$$$48\!\cdots\!24$$$$p^{18} T^{7} + 2168537688075866253 p^{27} T^{8} - 108420934 p^{36} T^{9} + p^{45} T^{10} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$