# Properties

 Label 24-384e12-1.1-c7e12-0-0 Degree $24$ Conductor $1.028\times 10^{31}$ Sign $1$ Analytic cond. $8.87681\times 10^{24}$ Root an. cond. $10.9524$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4.37e3·9-s − 8.30e4·17-s + 4.04e5·25-s − 9.69e5·41-s − 7.48e6·49-s + 1.80e7·73-s + 1.11e7·81-s + 4.35e7·89-s − 4.84e7·97-s − 8.76e7·113-s + 1.52e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3.63e8·153-s + 157-s + 163-s + 167-s + 4.45e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2·9-s − 4.10·17-s + 5.18·25-s − 2.19·41-s − 9.08·49-s + 5.43·73-s + 7/3·81-s + 6.55·89-s − 5.39·97-s − 5.71·113-s + 7.80·121-s + 8.20·153-s + 7.10·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$24$$ Conductor: $$2^{84} \cdot 3^{12}$$ Sign: $1$ Analytic conductor: $$8.87681\times 10^{24}$$ Root analytic conductor: $$10.9524$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.7955399625$$ $$L(\frac12)$$ $$\approx$$ $$0.7955399625$$ $$L(\frac{9}{2})$$ 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^{6} T^{2} )^{6}$$
good5 $$( 1 - 202422 T^{2} + 150456147 p^{3} T^{4} - 2272723482452 p^{4} T^{6} + 150456147 p^{17} T^{8} - 202422 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
7 $$( 1 + 3742386 T^{2} + 6535894742751 T^{4} + 6805623648777266172 T^{6} + 6535894742751 p^{14} T^{8} + 3742386 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
11 $$( 1 - 76035810 T^{2} + 2907631859152071 T^{4} -$$$$69\!\cdots\!84$$$$T^{6} + 2907631859152071 p^{14} T^{8} - 76035810 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
13 $$( 1 - 222990078 T^{2} + 20968007553597495 T^{4} -$$$$13\!\cdots\!60$$$$T^{6} + 20968007553597495 p^{14} T^{8} - 222990078 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
17 $$( 1 + 1222 p T + 710809823 T^{2} + 17058319559700 T^{3} + 710809823 p^{7} T^{4} + 1222 p^{15} T^{5} + p^{21} T^{6} )^{4}$$
19 $$( 1 - 1924649010 T^{2} + 2618941304588675895 T^{4} -$$$$26\!\cdots\!52$$$$T^{6} + 2618941304588675895 p^{14} T^{8} - 1924649010 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
23 $$( 1 + 8324910090 T^{2} + 43626867506508510495 T^{4} +$$$$16\!\cdots\!04$$$$T^{6} + 43626867506508510495 p^{14} T^{8} + 8324910090 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
29 $$( 1 - 74216131110 T^{2} +$$$$26\!\cdots\!43$$$$T^{4} -$$$$58\!\cdots\!20$$$$T^{6} +$$$$26\!\cdots\!43$$$$p^{14} T^{8} - 74216131110 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
31 $$( 1 + 92397120738 T^{2} +$$$$40\!\cdots\!19$$$$T^{4} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$40\!\cdots\!19$$$$p^{14} T^{8} + 92397120738 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
37 $$( 1 - 301868135502 T^{2} +$$$$54\!\cdots\!67$$$$T^{4} -$$$$61\!\cdots\!40$$$$T^{6} +$$$$54\!\cdots\!67$$$$p^{14} T^{8} - 301868135502 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
41 $$( 1 + 242258 T + 157515778679 T^{2} + 120350837328774876 T^{3} + 157515778679 p^{7} T^{4} + 242258 p^{14} T^{5} + p^{21} T^{6} )^{4}$$
43 $$( 1 - 1139303049378 T^{2} +$$$$57\!\cdots\!27$$$$T^{4} -$$$$18\!\cdots\!44$$$$T^{6} +$$$$57\!\cdots\!27$$$$p^{14} T^{8} - 1139303049378 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
47 $$( 1 + 2213950484346 T^{2} +$$$$22\!\cdots\!87$$$$T^{4} +$$$$13\!\cdots\!96$$$$T^{6} +$$$$22\!\cdots\!87$$$$p^{14} T^{8} + 2213950484346 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
53 $$( 1 - 4140796683414 T^{2} +$$$$98\!\cdots\!47$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$98\!\cdots\!47$$$$p^{14} T^{8} - 4140796683414 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
59 $$( 1 - 9172921230786 T^{2} +$$$$45\!\cdots\!15$$$$T^{4} -$$$$13\!\cdots\!20$$$$T^{6} +$$$$45\!\cdots\!15$$$$p^{14} T^{8} - 9172921230786 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
61 $$( 1 - 11980240333758 T^{2} +$$$$75\!\cdots\!03$$$$T^{4} -$$$$29\!\cdots\!92$$$$T^{6} +$$$$75\!\cdots\!03$$$$p^{14} T^{8} - 11980240333758 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
67 $$( 1 - 25642072420626 T^{2} +$$$$47\!\cdots\!37$$$$p T^{4} -$$$$24\!\cdots\!96$$$$T^{6} +$$$$47\!\cdots\!37$$$$p^{15} T^{8} - 25642072420626 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
71 $$( 1 + 5015239048170 T^{2} +$$$$99\!\cdots\!43$$$$T^{4} +$$$$12\!\cdots\!40$$$$T^{6} +$$$$99\!\cdots\!43$$$$p^{14} T^{8} + 5015239048170 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
73 $$( 1 - 4519914 T + 9068728164087 T^{2} - 3826246015633904620 T^{3} + 9068728164087 p^{7} T^{4} - 4519914 p^{14} T^{5} + p^{21} T^{6} )^{4}$$
79 $$( 1 + 69274085093826 T^{2} +$$$$26\!\cdots\!03$$$$T^{4} +$$$$61\!\cdots\!12$$$$T^{6} +$$$$26\!\cdots\!03$$$$p^{14} T^{8} + 69274085093826 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
83 $$( 1 - 40550002231986 T^{2} +$$$$18\!\cdots\!91$$$$T^{4} -$$$$50\!\cdots\!28$$$$T^{6} +$$$$18\!\cdots\!91$$$$p^{14} T^{8} - 40550002231986 p^{28} T^{10} + p^{42} T^{12} )^{2}$$
89 $$( 1 - 10890786 T + 89854806381351 T^{2} -$$$$74\!\cdots\!92$$$$T^{3} + 89854806381351 p^{7} T^{4} - 10890786 p^{14} T^{5} + p^{21} T^{6} )^{4}$$
97 $$( 1 + 12114582 T + 207440378530095 T^{2} +$$$$17\!\cdots\!24$$$$T^{3} + 207440378530095 p^{7} T^{4} + 12114582 p^{14} T^{5} + p^{21} T^{6} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$