# Properties

 Label 16-12e24-1.1-c3e8-0-8 Degree $16$ Conductor $7.950\times 10^{25}$ Sign $1$ Analytic cond. $1.16755\times 10^{16}$ Root an. cond. $10.0972$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 40·25-s + 2.06e3·49-s + 3.12e3·73-s − 3.41e3·97-s + 9.68e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.55e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
 L(s)  = 1 − 0.319·25-s + 6.02·49-s + 5.01·73-s − 3.57·97-s + 7.27·121-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 + 7.07·169-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 + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-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(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{48} \cdot 3^{24}$$ Sign: $1$ Analytic conductor: $$1.16755\times 10^{16}$$ Root analytic conductor: $$10.0972$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$60.74302225$$ $$L(\frac12)$$ $$\approx$$ $$60.74302225$$ $$L(\frac{5}{2})$$ 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$$
good5 $$( 1 + 2 p T^{2} + p^{6} T^{4} )^{4}$$
7 $$( 1 - 517 T^{2} + p^{6} T^{4} )^{4}$$
11 $$( 1 - 2422 T^{2} + p^{6} T^{4} )^{4}$$
13 $$( 1 - 7 p T + p^{3} T^{2} )^{4}( 1 + 7 p T + p^{3} T^{2} )^{4}$$
17 $$( 1 - 9106 T^{2} + p^{6} T^{4} )^{4}$$
19 $$( 1 + 7091 T^{2} + p^{6} T^{4} )^{4}$$
23 $$( 1 + 23614 T^{2} + p^{6} T^{4} )^{4}$$
29 $$( 1 + 14218 T^{2} + p^{6} T^{4} )^{4}$$
31 $$( 1 - 29998 T^{2} + p^{6} T^{4} )^{4}$$
37 $$( 1 - 100223 T^{2} + p^{6} T^{4} )^{4}$$
41 $$( 1 - 65842 T^{2} + p^{6} T^{4} )^{4}$$
43 $$( 1 + 120026 T^{2} + p^{6} T^{4} )^{4}$$
47 $$( 1 + 120526 T^{2} + p^{6} T^{4} )^{4}$$
53 $$( 1 + 20314 T^{2} + p^{6} T^{4} )^{4}$$
59 $$( 1 + 75242 T^{2} + p^{6} T^{4} )^{4}$$
61 $$( 1 - 233639 T^{2} + p^{6} T^{4} )^{4}$$
67 $$( 1 + 601163 T^{2} + p^{6} T^{4} )^{4}$$
71 $$( 1 + 367342 T^{2} + p^{6} T^{4} )^{4}$$
73 $$( 1 - 391 T + p^{3} T^{2} )^{8}$$
79 $$( 1 - 815509 T^{2} + p^{6} T^{4} )^{4}$$
83 $$( 1 - 1128214 T^{2} + p^{6} T^{4} )^{4}$$
89 $$( 1 + 318782 T^{2} + p^{6} T^{4} )^{4}$$
97 $$( 1 + 427 T + p^{3} T^{2} )^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$