# Properties

 Label 16-4608e8-1.1-c1e8-0-16 Degree $16$ Conductor $2.033\times 10^{29}$ Sign $1$ Analytic cond. $3.35982\times 10^{12}$ Root an. cond. $6.06589$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·5-s + 8·13-s + 32·25-s + 8·29-s + 40·37-s + 24·49-s − 8·53-s + 40·61-s + 64·65-s + 64·97-s − 40·101-s − 40·109-s − 16·113-s + 88·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 3.57·5-s + 2.21·13-s + 32/5·25-s + 1.48·29-s + 6.57·37-s + 24/7·49-s − 1.09·53-s + 5.12·61-s + 7.93·65-s + 6.49·97-s − 3.98·101-s − 3.83·109-s − 1.50·113-s + 7.87·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{72} \cdot 3^{16}$$ Sign: $1$ Analytic conductor: $$3.35982\times 10^{12}$$ Root analytic conductor: $$6.06589$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{4608} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{72} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$186.3933070$$ $$L(\frac12)$$ $$\approx$$ $$186.3933070$$ $$L(\frac{3}{2})$$ 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$$
good5 $$( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
7 $$( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
11 $$1 - 460 T^{4} + 82054 T^{8} - 460 p^{4} T^{12} + p^{8} T^{16}$$
13 $$( 1 - 4 T + 8 T^{2} - 44 T^{3} + 238 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
17 $$( 1 + 26 T^{2} + p^{2} T^{4} )^{4}$$
19 $$1 - 12 T^{4} - 247354 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16}$$
23 $$( 1 - 12 T^{2} + 1062 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
29 $$( 1 - 4 T + 8 T^{2} + 20 T^{3} - 1106 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
37 $$( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 11662 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
41 $$( 1 - 66 T^{2} + p^{2} T^{4} )^{4}$$
43 $$1 + 6068 T^{4} + 15827590 T^{8} + 6068 p^{4} T^{12} + p^{8} T^{16}$$
47 $$( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
53 $$( 1 + 4 T + 8 T^{2} + 76 T^{3} - 434 T^{4} + 76 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
59 $$1 + 12084 T^{4} + 60324614 T^{8} + 12084 p^{4} T^{12} + p^{8} T^{16}$$
61 $$( 1 - 20 T + 200 T^{2} - 1500 T^{3} + 11054 T^{4} - 1500 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 + 116 T^{4} - 21877946 T^{8} + 116 p^{4} T^{12} + p^{8} T^{16}$$
71 $$( 1 - 76 T^{2} + 9958 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
79 $$( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
83 $$1 - 3468 T^{4} - 36032314 T^{8} - 3468 p^{4} T^{12} + p^{8} T^{16}$$
89 $$( 1 - 332 T^{2} + 43270 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$