# Properties

 Label 16-300e8-1.1-c1e8-0-0 Degree $16$ Conductor $6.561\times 10^{19}$ Sign $1$ Analytic cond. $1084.39$ Root an. cond. $1.54774$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 4·16-s − 16·19-s − 16·41-s + 64·59-s + 24·61-s + 16·64-s + 64·76-s − 32·79-s − 2·81-s + 16·101-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 64·164-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2·4-s + 16-s − 3.67·19-s − 2.49·41-s + 8.33·59-s + 3.07·61-s + 2·64-s + 7.34·76-s − 3.60·79-s − 2/9·81-s + 1.59·101-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 4.99·164-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 3^{8} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$1084.39$$ Root analytic conductor: $$1.54774$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{300} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.05940581789$$ $$L(\frac12)$$ $$\approx$$ $$0.05940581789$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
3 $$( 1 + T^{4} )^{2}$$
5 $$1$$
good7 $$1 - 2 p^{2} T^{4} + 7011 T^{8} - 2 p^{6} T^{12} + p^{8} T^{16}$$
11 $$( 1 - 18 T^{2} + p^{2} T^{4} )^{4}$$
13 $$1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16}$$
17 $$( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
19 $$( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4}$$
23 $$1 + 68 T^{4} - 434490 T^{8} + 68 p^{4} T^{12} + p^{8} T^{16}$$
29 $$( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
31 $$( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
37 $$1 + 868 T^{4} + 2707878 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16}$$
41 $$( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4}$$
43 $$1 + 3358 T^{4} + 8633475 T^{8} + 3358 p^{4} T^{12} + p^{8} T^{16}$$
47 $$1 + 5188 T^{4} + 13723398 T^{8} + 5188 p^{4} T^{12} + p^{8} T^{16}$$
53 $$1 + 5348 T^{4} + 16710438 T^{8} + 5348 p^{4} T^{12} + p^{8} T^{16}$$
59 $$( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4}$$
61 $$( 1 - 3 T + p T^{2} )^{8}$$
67 $$1 - 6818 T^{4} + 25780611 T^{8} - 6818 p^{4} T^{12} + p^{8} T^{16}$$
71 $$( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 + 6242 T^{4} + p^{4} T^{8} )^{2}$$
79 $$( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4}$$
83 $$1 - 284 T^{4} - 90968346 T^{8} - 284 p^{4} T^{12} + p^{8} T^{16}$$
89 $$( 1 - 130 T^{2} + p^{2} T^{4} )^{4}$$
97 $$1 - 3842 T^{4} + 131200515 T^{8} - 3842 p^{4} T^{12} + p^{8} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$