# Properties

 Label 16-48e16-1.1-c2e8-0-8 Degree $16$ Conductor $7.941\times 10^{26}$ Sign $1$ Analytic cond. $2.41290\times 10^{14}$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 80·25-s + 128·31-s − 184·49-s + 160·73-s + 384·79-s − 192·97-s + 896·103-s + 360·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 408·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 + 16/5·25-s + 4.12·31-s − 3.75·49-s + 2.19·73-s + 4.86·79-s − 1.97·97-s + 8.69·103-s + 2.97·121-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 − 2.41·169-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 + 0.00448·223-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{64} \cdot 3^{16}$$ Sign: $1$ Analytic conductor: $$2.41290\times 10^{14}$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2304} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$33.64923390$$ $$L(\frac12)$$ $$\approx$$ $$33.64923390$$ $$L(2)$$ 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 - 8 p T^{2} + 818 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} )^{2}$$
7 $$( 1 + 46 T^{2} + p^{4} T^{4} )^{4}$$
11 $$( 1 - 180 T^{2} + 24070 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 + 204 T^{2} + 14278 T^{4} + 204 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
17 $$( 1 - 320 T^{2} + 189314 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
19 $$( 1 - 60 T^{2} + 48550 T^{4} - 60 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
23 $$( 1 - 1652 T^{2} + 1188710 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
29 $$( 1 - 2376 T^{2} + 2685298 T^{4} - 2376 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
31 $$( 1 - 32 T + 1710 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
37 $$( 1 + 3972 T^{2} + 7479526 T^{4} + 3972 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
41 $$( 1 - 5792 T^{2} + 13955138 T^{4} - 5792 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
43 $$( 1 + 996 T^{2} + 6872614 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
47 $$( 1 - 4660 T^{2} + 10875174 T^{4} - 4660 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 8712 T^{2} + 34754866 T^{4} - 8712 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$( 1 - 9316 T^{2} + 45079718 T^{4} - 9316 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
61 $$( 1 + 9412 T^{2} + 45525030 T^{4} + 9412 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
67 $$( 1 + 12484 T^{2} + 74951718 T^{4} + 12484 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
71 $$( 1 - 9076 T^{2} + 54164454 T^{4} - 9076 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 40 T + 7730 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
79 $$( 1 - 96 T + 8494 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
83 $$( 1 - 24436 T^{2} + 241946438 T^{4} - 24436 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
89 $$( 1 - 25856 T^{2} + 284297666 T^{4} - 25856 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
97 $$( 1 + 48 T + 18562 T^{2} + 48 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}$$