# Properties

 Label 16-384e8-1.1-c2e8-0-3 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Analytic cond. $1.43658\times 10^{8}$ Root an. cond. $3.23469$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·5-s − 12·9-s + 48·13-s + 16·17-s + 24·25-s − 80·29-s − 16·37-s + 80·41-s + 192·45-s + 152·49-s + 176·53-s − 272·61-s − 768·65-s − 16·73-s + 90·81-s − 256·85-s − 240·89-s + 400·97-s − 528·101-s + 560·109-s + 336·113-s − 576·117-s + 520·121-s + 1.04e3·125-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 3.19·5-s − 4/3·9-s + 3.69·13-s + 0.941·17-s + 0.959·25-s − 2.75·29-s − 0.432·37-s + 1.95·41-s + 4.26·45-s + 3.10·49-s + 3.32·53-s − 4.45·61-s − 11.8·65-s − 0.219·73-s + 10/9·81-s − 3.01·85-s − 2.69·89-s + 4.12·97-s − 5.22·101-s + 5.13·109-s + 2.97·113-s − 4.92·117-s + 4.29·121-s + 8.31·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$1.43658\times 10^{8}$$ Root analytic conductor: $$3.23469$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.4222018189$$ $$L(\frac12)$$ $$\approx$$ $$0.4222018189$$ $$L(2)$$ 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 T^{2} )^{4}$$
good5 $$( 1 + 8 T + 84 T^{2} + 472 T^{3} + 2822 T^{4} + 472 p^{2} T^{5} + 84 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
7 $$1 - 152 T^{2} + 1476 p T^{4} - 514600 T^{6} + 25187654 T^{8} - 514600 p^{4} T^{10} + 1476 p^{9} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16}$$
11 $$( 1 - 260 T^{2} + 40038 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 - 24 T + 540 T^{2} - 5736 T^{3} + 89702 T^{4} - 5736 p^{2} T^{5} + 540 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
17 $$( 1 - 8 T + 252 T^{2} + 2888 T^{3} + 48902 T^{4} + 2888 p^{2} T^{5} + 252 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
19 $$1 - 1224 T^{2} + 1027804 T^{4} - 554589432 T^{6} + 236077316358 T^{8} - 554589432 p^{4} T^{10} + 1027804 p^{8} T^{12} - 1224 p^{12} T^{14} + p^{16} T^{16}$$
23 $$1 - 3272 T^{2} + 5007132 T^{4} - 4715977336 T^{6} + 5669682278 p^{2} T^{8} - 4715977336 p^{4} T^{10} + 5007132 p^{8} T^{12} - 3272 p^{12} T^{14} + p^{16} T^{16}$$
29 $$( 1 + 40 T + 2196 T^{2} + 31160 T^{3} + 1501766 T^{4} + 31160 p^{2} T^{5} + 2196 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
31 $$1 - 4248 T^{2} + 9652444 T^{4} - 14917063464 T^{6} + 16677690731718 T^{8} - 14917063464 p^{4} T^{10} + 9652444 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16}$$
37 $$( 1 + 8 T + 1980 T^{2} - 54920 T^{3} + 1119206 T^{4} - 54920 p^{2} T^{5} + 1980 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
41 $$( 1 - 40 T + 2556 T^{2} + 43240 T^{3} - 289594 T^{4} + 43240 p^{2} T^{5} + 2556 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
43 $$1 - 5960 T^{2} + 23041116 T^{4} - 59814611320 T^{6} + 126481252740230 T^{8} - 59814611320 p^{4} T^{10} + 23041116 p^{8} T^{12} - 5960 p^{12} T^{14} + p^{16} T^{16}$$
47 $$1 - 12360 T^{2} + 70380700 T^{4} - 250403346936 T^{6} + 638353123484742 T^{8} - 250403346936 p^{4} T^{10} + 70380700 p^{8} T^{12} - 12360 p^{12} T^{14} + p^{16} T^{16}$$
53 $$( 1 - 88 T + 10068 T^{2} - 520520 T^{3} + 37632134 T^{4} - 520520 p^{2} T^{5} + 10068 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
59 $$1 - 7688 T^{2} + 37931484 T^{4} - 124184601784 T^{6} + 382130398823558 T^{8} - 124184601784 p^{4} T^{10} + 37931484 p^{8} T^{12} - 7688 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 + 136 T + 14076 T^{2} + 1001336 T^{3} + 63819302 T^{4} + 1001336 p^{2} T^{5} + 14076 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
67 $$( 1 - 8644 T^{2} + 58097190 T^{4} - 8644 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
71 $$1 - 18120 T^{2} + 152088604 T^{4} - 903092412024 T^{6} + 4737261672397254 T^{8} - 903092412024 p^{4} T^{10} + 152088604 p^{8} T^{12} - 18120 p^{12} T^{14} + p^{16} T^{16}$$
73 $$( 1 + 8 T + 5212 T^{2} + 653240 T^{3} - 3523322 T^{4} + 653240 p^{2} T^{5} + 5212 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
79 $$1 - 26776 T^{2} + 333200860 T^{4} - 2815930225960 T^{6} + 19175645821896646 T^{8} - 2815930225960 p^{4} T^{10} + 333200860 p^{8} T^{12} - 26776 p^{12} T^{14} + p^{16} T^{16}$$
83 $$1 - 15752 T^{2} + 70662492 T^{4} - 102795674680 T^{6} + 590797586843654 T^{8} - 102795674680 p^{4} T^{10} + 70662492 p^{8} T^{12} - 15752 p^{12} T^{14} + p^{16} T^{16}$$
89 $$( 1 + 120 T + 30300 T^{2} + 2699976 T^{3} + 352873862 T^{4} + 2699976 p^{2} T^{5} + 30300 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
97 $$( 1 - 200 T + 21276 T^{2} - 3009400 T^{3} + 385130822 T^{4} - 3009400 p^{2} T^{5} + 21276 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$