# Properties

 Label 16-12e24-1.1-c1e8-0-6 Degree $16$ Conductor $7.950\times 10^{25}$ Sign $1$ Analytic cond. $1.31391\times 10^{9}$ Root an. cond. $3.71458$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·13-s + 20·25-s + 28·49-s − 32·61-s − 24·73-s − 40·97-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
 L(s)  = 1 + 4.43·13-s + 4·25-s + 4·49-s − 4.09·61-s − 2.80·73-s − 4.06·97-s − 4·121-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 + 0.0773·167-s + 10.4·169-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 + 0.0663·227-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\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 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{48} \cdot 3^{24}$$ Sign: $1$ Analytic conductor: $$1.31391\times 10^{9}$$ Root analytic conductor: $$3.71458$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$20.96975914$$ $$L(\frac12)$$ $$\approx$$ $$20.96975914$$ $$L(\frac{3}{2})$$ 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 - 2 p T^{2} + 51 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2}$$
7 $$( 1 - 2 p T^{2} + 123 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2}$$
11 $$( 1 + 2 p T^{2} + 267 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2}$$
13 $$( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4}$$
17 $$( 1 - 22 T^{2} + p^{2} T^{4} )^{4}$$
19 $$( 1 - 14 T^{2} + p^{2} T^{4} )^{4}$$
23 $$( 1 + 38 T^{2} + p^{2} T^{4} )^{4}$$
29 $$( 1 - 28 T^{2} + 342 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
31 $$( 1 - 2 p T^{2} + 2283 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2}$$
37 $$( 1 + 50 T^{2} + p^{2} T^{4} )^{4}$$
41 $$( 1 - 52 T^{2} + 2502 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
43 $$( 1 - 116 T^{2} + 6678 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
47 $$( 1 + 100 T^{2} + 5382 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
53 $$( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
59 $$( 1 + 148 T^{2} + 10902 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$( 1 + 4 T + p T^{2} )^{8}$$
67 $$( 1 - 20 T^{2} - 522 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
71 $$( 1 + 52 T^{2} + 7302 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}$$
79 $$( 1 - 154 T^{2} + p^{2} T^{4} )^{4}$$
83 $$( 1 - 74 T^{2} + 12747 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 - 188 T^{2} + 21222 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 + 5 T + p T^{2} )^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$