# Properties

 Label 16-12e24-1.1-c3e8-0-7 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 − 1.22e3·23-s − 496·25-s + 1.80e3·47-s + 1.25e3·49-s − 432·71-s + 344·73-s + 1.24e3·97-s + 3.80e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.31e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 − 11.0·23-s − 3.96·25-s + 5.58·47-s + 3.65·49-s − 0.722·71-s + 0.551·73-s + 1.29·97-s + 2.86·121-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 + 5.98·169-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 + ⋯

## 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

 $$L(2)$$ $$\approx$$ $$5.310147773$$ $$L(\frac12)$$ $$\approx$$ $$5.310147773$$ $$L(\frac{5}{2})$$ 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 + 248 T^{2} + 1806 p^{2} T^{4} + 248 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
7 $$( 1 - 626 T^{2} + 327363 T^{4} - 626 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
11 $$( 1 - 1904 T^{2} + 3668622 T^{4} - 1904 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
13 $$( 1 - 3287 T^{2} + p^{6} T^{4} )^{4}$$
17 $$( 1 - 5072 T^{2} + 1982238 T^{4} - 5072 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
19 $$( 1 + 18094 T^{2} + 171635955 T^{4} + 18094 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
23 $$( 1 + 306 T + 44422 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
29 $$( 1 + 41866 T^{2} + p^{6} T^{4} )^{4}$$
31 $$( 1 - 116204 T^{2} + 5150826150 T^{4} - 116204 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
37 $$( 1 - 92182 T^{2} + 7044936315 T^{4} - 92182 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
41 $$( 1 - 30740 T^{2} + 8341944198 T^{4} - 30740 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
43 $$( 1 + 65524 T^{2} + 6829649142 T^{4} + 65524 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
47 $$( 1 - 450 T + 95542 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
53 $$( 1 + 154436 T^{2} + 4597192086 T^{4} + 154436 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
59 $$( 1 - 512528 T^{2} + 126780934542 T^{4} - 512528 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
61 $$( 1 - 763582 T^{2} + 246326003187 T^{4} - 763582 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
67 $$( 1 + 743782 T^{2} + 274440697755 T^{4} + 743782 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
71 $$( 1 + 108 T + 599182 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
73 $$( 1 - 86 T + 65499 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
79 $$( 1 - 1357322 T^{2} + 905419894299 T^{4} - 1357322 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
83 $$( 1 - 1580396 T^{2} + 1168518522966 T^{4} - 1580396 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
89 $$( 1 - 2136560 T^{2} + 2018716991358 T^{4} - 2136560 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
97 $$( 1 - 310 T + 926871 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$