# Properties

 Label 16-1216e8-1.1-c1e8-0-2 Degree $16$ Conductor $4.780\times 10^{24}$ Sign $1$ Analytic cond. $7.90106\times 10^{7}$ Root an. cond. $3.11605$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·9-s − 24·17-s − 8·25-s + 52·49-s + 56·73-s − 26·81-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 96·153-s + 157-s + 163-s + 167-s − 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 − 4/3·9-s − 5.82·17-s − 8/5·25-s + 52/7·49-s + 6.55·73-s − 2.88·81-s − 8·121-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 + 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{48} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$7.90106\times 10^{7}$$ Root analytic conductor: $$3.11605$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1216} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5045398401$$ $$L(\frac12)$$ $$\approx$$ $$0.5045398401$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
good3 $$( 1 + T^{2} + p^{2} T^{4} )^{4}$$
5 $$( 1 + 2 T^{2} + p^{2} T^{4} )^{4}$$
7 $$( 1 - 13 T^{2} + p^{2} T^{4} )^{4}$$
11 $$( 1 + p T^{2} )^{8}$$
13 $$( 1 + 19 T^{2} + p^{2} T^{4} )^{4}$$
17 $$( 1 + 3 T + p T^{2} )^{8}$$
23 $$( 1 - 37 T^{2} + p^{2} T^{4} )^{4}$$
29 $$( 1 - 5 T^{2} + p^{2} T^{4} )^{4}$$
31 $$( 1 + p T^{2} )^{8}$$
37 $$( 1 - 38 T^{2} + p^{2} T^{4} )^{4}$$
41 $$( 1 + 2 T^{2} + p^{2} T^{4} )^{4}$$
43 $$( 1 - 22 T^{2} + p^{2} T^{4} )^{4}$$
47 $$( 1 - p T^{2} )^{8}$$
53 $$( 1 + 43 T^{2} + p^{2} T^{4} )^{4}$$
59 $$( 1 - 55 T^{2} + p^{2} T^{4} )^{4}$$
61 $$( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4}$$
67 $$( 1 + 41 T^{2} + p^{2} T^{4} )^{4}$$
71 $$( 1 + 58 T^{2} + p^{2} T^{4} )^{4}$$
73 $$( 1 - 7 T + p T^{2} )^{8}$$
79 $$( 1 + 74 T^{2} + p^{2} T^{4} )^{4}$$
83 $$( 1 - 134 T^{2} + p^{2} T^{4} )^{4}$$
89 $$( 1 - 94 T^{2} + p^{2} T^{4} )^{4}$$
97 $$( 1 - 110 T^{2} + p^{2} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$