# Properties

 Label 16-384e8-1.1-c6e8-0-2 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Analytic cond. $3.70927\times 10^{15}$ Root an. cond. $9.39897$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 972·9-s + 4.46e3·17-s + 1.23e5·25-s − 9.89e4·41-s + 3.31e5·49-s + 7.69e4·73-s + 5.90e5·81-s + 6.63e6·89-s − 4.92e6·97-s − 3.15e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.33e6·153-s + 157-s + 163-s + 167-s + 1.76e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 + 4/3·9-s + 0.908·17-s + 7.89·25-s − 1.43·41-s + 2.81·49-s + 0.197·73-s + 10/9·81-s + 9.41·89-s − 5.40·97-s − 1.77·121-s + 1.21·153-s + 3.65·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$3.70927\times 10^{15}$$ Root analytic conductor: $$9.39897$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$53.27559381$$ $$L(\frac12)$$ $$\approx$$ $$53.27559381$$ $$L(4)$$ 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^{5} T^{2} )^{4}$$
good5 $$( 1 - 1234 p^{2} T^{2} + p^{12} T^{4} )^{4}$$
7 $$( 1 - 165604 T^{2} + 18091945734 T^{4} - 165604 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
11 $$( 1 + 1576516 T^{2} + 319487479206 T^{4} + 1576516 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
13 $$( 1 - 8814820 T^{2} + 39706553357862 T^{4} - 8814820 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
17 $$( 1 - 1116 T - 2175226 T^{2} - 1116 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
19 $$( 1 + 154272388 T^{2} + 10374253923756390 T^{4} + 154272388 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
23 $$( 1 - 279579460 T^{2} + 41996156363862342 T^{4} - 279579460 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
29 $$( 1 - 3297092 T^{2} + 49459707520391526 T^{4} - 3297092 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
31 $$( 1 - 2377420708 T^{2} + 2864250173428593990 T^{4} - 2377420708 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
37 $$( 1 - 4472699044 T^{2} + 17273824012262262246 T^{4} - 4472699044 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
41 $$( 1 + 24732 T + 9196270886 T^{2} + 24732 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
43 $$( 1 + 14783863108 T^{2} +$$$$11\!\cdots\!06$$$$T^{4} + 14783863108 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
47 $$( 1 - 11965054468 T^{2} + 35254807719429905286 T^{4} - 11965054468 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
53 $$( 1 - 29043574916 T^{2} +$$$$77\!\cdots\!46$$$$T^{4} - 29043574916 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
59 $$( 1 + 127672933828 T^{2} +$$$$72\!\cdots\!58$$$$T^{4} + 127672933828 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
61 $$( 1 - 196576229092 T^{2} +$$$$14\!\cdots\!66$$$$T^{4} - 196576229092 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
67 $$( 1 - 65276716028 T^{2} +$$$$16\!\cdots\!10$$$$T^{4} - 65276716028 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
71 $$( 1 - 306730507972 T^{2} +$$$$46\!\cdots\!66$$$$T^{4} - 306730507972 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
73 $$( 1 - 19228 T + 154739682726 T^{2} - 19228 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
79 $$( 1 - 374724961060 T^{2} +$$$$15\!\cdots\!82$$$$T^{4} - 374724961060 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
83 $$( 1 + 849400576132 T^{2} +$$$$37\!\cdots\!78$$$$T^{4} + 849400576132 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
89 $$( 1 - 1659708 T + 1677544070438 T^{2} - 1659708 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
97 $$( 1 + 1232420 T + 1634488777158 T^{2} + 1232420 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$