# Properties

 Label 16-384e8-1.1-c3e8-0-2 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Analytic cond. $6.94349\times 10^{10}$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 60·9-s − 640·23-s − 392·25-s + 984·49-s − 640·71-s − 1.23e3·73-s + 1.24e3·81-s + 5.71e3·97-s + 6.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.06e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 3.84e4·207-s + ⋯
 L(s)  = 1 − 2.22·9-s − 5.80·23-s − 3.13·25-s + 2.86·49-s − 1.06·71-s − 1.97·73-s + 1.70·81-s + 5.97·97-s + 4.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.12·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 12.8·207-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(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$6.94349\times 10^{10}$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.339300794$$ $$L(\frac12)$$ $$\approx$$ $$2.339300794$$ $$L(\frac{5}{2})$$ 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 + 10 p T^{2} + p^{6} T^{4} )^{2}$$
good5 $$( 1 + 196 T^{2} + 774 p^{2} T^{4} + 196 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
7 $$( 1 - 492 T^{2} + 102278 T^{4} - 492 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
11 $$( 1 - 3108 T^{2} + 5613974 T^{4} - 3108 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
13 $$( 1 - 4532 T^{2} + 10573590 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
17 $$( 1 - 6916 T^{2} + 54727878 T^{4} - 6916 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
19 $$( 1 - 3588 T^{2} + 29873654 T^{4} - 3588 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
23 $$( 1 + 160 T + 29390 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
29 $$( 1 + 28900 T^{2} + 841043958 T^{4} + 28900 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
31 $$( 1 - 112780 T^{2} + 4945356198 T^{4} - 112780 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
37 $$( 1 - 40148 T^{2} + 1484010870 T^{4} - 40148 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
41 $$( 1 - 198884 T^{2} + 18596196390 T^{4} - 198884 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
43 $$( 1 + 174108 T^{2} + 16916734358 T^{4} + 174108 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
47 $$( 1 + 121630 T^{2} + p^{6} T^{4} )^{4}$$
53 $$( 1 + 133508 T^{2} - 4558541226 T^{4} + 133508 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
59 $$( 1 - 408658 T^{2} + p^{6} T^{4} )^{4}$$
61 $$( 1 - 129524 T^{2} + 106016792790 T^{4} - 129524 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
67 $$( 1 + 690876 T^{2} + 254657072438 T^{4} + 690876 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
71 $$( 1 + 160 T - 117778 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
73 $$( 1 + 308 T + 608214 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
79 $$( 1 - 257100 T^{2} + 501931869158 T^{4} - 257100 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
83 $$( 1 - 1153988 T^{2} + 779081408118 T^{4} - 1153988 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
89 $$( 1 - 1957732 T^{2} + 1892598888294 T^{4} - 1957732 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
97 $$( 1 - 1428 T + 2249126 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$