# Properties

 Label 16-864e8-1.1-c2e8-0-7 Degree $16$ Conductor $3.105\times 10^{23}$ Sign $1$ Analytic cond. $9.43605\times 10^{10}$ Root an. cond. $4.85204$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·13-s + 84·25-s + 32·37-s − 148·49-s + 96·61-s − 216·73-s − 40·97-s + 352·109-s + 900·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 920·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 1.23·13-s + 3.35·25-s + 0.864·37-s − 3.02·49-s + 1.57·61-s − 2.95·73-s − 0.412·97-s + 3.22·109-s + 7.43·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.44·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{40} \cdot 3^{24}$$ Sign: $1$ Analytic conductor: $$9.43605\times 10^{10}$$ Root analytic conductor: $$4.85204$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{40} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.744508283$$ $$L(\frac12)$$ $$\approx$$ $$3.744508283$$ $$L(2)$$ 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 - 42 T^{2} + 1043 T^{4} - 42 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
7 $$( 1 + 74 T^{2} + 2643 T^{4} + 74 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
11 $$( 1 - 450 T^{2} + 79619 T^{4} - 450 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 + 4 T + 270 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
17 $$( 1 - 76 T^{2} + 127014 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
19 $$( 1 + 980 T^{2} + 459270 T^{4} + 980 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
23 $$( 1 - 1684 T^{2} + 1227174 T^{4} - 1684 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
29 $$( 1 - 2028 T^{2} + 2147846 T^{4} - 2028 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
31 $$( 1 - 646 T^{2} - 355581 T^{4} - 646 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
37 $$( 1 - 8 T + 954 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
41 $$( 1 - 5076 T^{2} + 12019238 T^{4} - 5076 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
43 $$( 1 + 1916 T^{2} + 7662054 T^{4} + 1916 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
47 $$( 1 - 5052 T^{2} + 14107910 T^{4} - 5052 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 2538 T^{2} + 14597075 T^{4} - 2538 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$( 1 - 6492 T^{2} + 25440038 T^{4} - 6492 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
61 $$( 1 - 24 T + 2978 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
67 $$( 1 + 7676 T^{2} + 45068838 T^{4} + 7676 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
71 $$( 1 - 5292 T^{2} + 24922406 T^{4} - 5292 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 + 54 T + 2675 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
79 $$( 1 + 11660 T^{2} + 68532390 T^{4} + 11660 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
83 $$( 1 - 3634 T^{2} - 15003021 T^{4} - 3634 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
89 $$( 1 + 7332 T^{2} + 109063046 T^{4} + 7332 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
97 $$( 1 + 10 T + 14235 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$