# Properties

 Degree $16$ Conductor $7.985\times 10^{28}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·11-s − 12·19-s + 8·29-s − 16·31-s − 8·41-s + 12·49-s − 24·59-s + 48·61-s − 4·71-s + 36·79-s − 4·81-s − 8·89-s − 24·101-s − 40·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 2.41·11-s − 2.75·19-s + 1.48·29-s − 2.87·31-s − 1.24·41-s + 12/7·49-s − 3.12·59-s + 6.14·61-s − 0.474·71-s + 4.05·79-s − 4/9·81-s − 0.847·89-s − 2.38·101-s − 3.83·109-s − 0.363·121-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 + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 5^{16} \cdot 41^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{4100} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 5^{16} \cdot 41^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5404407845$$ $$L(\frac12)$$ $$\approx$$ $$0.5404407845$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
41 $$( 1 + T )^{8}$$
good3 $$1 + 4 T^{4} - 4 p^{2} T^{6} - 14 T^{8} - 4 p^{4} T^{10} + 4 p^{4} T^{12} + p^{8} T^{16}$$
7 $$1 - 12 T^{2} + 12 p T^{4} - 200 T^{6} - 558 T^{8} - 200 p^{2} T^{10} + 12 p^{5} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16}$$
11 $$( 1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
13 $$1 - 24 T^{2} + 380 T^{4} - 4584 T^{6} + 59814 T^{8} - 4584 p^{2} T^{10} + 380 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16}$$
17 $$1 - 24 T^{2} + 28 p T^{4} - 40 p T^{6} + 3398 T^{8} - 40 p^{3} T^{10} + 28 p^{5} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16}$$
19 $$( 1 + 6 T + 62 T^{2} + 208 T^{3} + 1448 T^{4} + 208 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
23 $$1 - 72 T^{2} + 100 p T^{4} - 40952 T^{6} + 679622 T^{8} - 40952 p^{2} T^{10} + 100 p^{5} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16}$$
29 $$( 1 - 4 T + 76 T^{2} - 12 p T^{3} + 2870 T^{4} - 12 p^{2} T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
37 $$1 - 168 T^{2} + 14516 T^{4} - 846120 T^{6} + 36244134 T^{8} - 846120 p^{2} T^{10} + 14516 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16}$$
43 $$1 - 232 T^{2} + 26780 T^{4} - 1971224 T^{6} + 100629414 T^{8} - 1971224 p^{2} T^{10} + 26780 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 - 216 T^{2} + 25412 T^{4} - 1955324 T^{6} + 108019538 T^{8} - 1955324 p^{2} T^{10} + 25412 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16}$$
53 $$1 - 168 T^{2} + 17692 T^{4} - 1367128 T^{6} + 81443238 T^{8} - 1367128 p^{2} T^{10} + 17692 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 1 + 12 T + 252 T^{2} + 1996 T^{3} + 22582 T^{4} + 1996 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
61 $$( 1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 - 292 T^{2} + 46132 T^{4} - 4949152 T^{6} + 385553458 T^{8} - 4949152 p^{2} T^{10} + 46132 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16}$$
71 $$( 1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$1 - 360 T^{2} + 67764 T^{4} - 8331176 T^{6} + 717581958 T^{8} - 8331176 p^{2} T^{10} + 67764 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 1 - 18 T + 366 T^{2} - 4224 T^{3} + 45328 T^{4} - 4224 p T^{5} + 366 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$1 - 360 T^{2} + 73244 T^{4} - 9891864 T^{6} + 961607526 T^{8} - 9891864 p^{2} T^{10} + 73244 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16}$$
89 $$( 1 + 4 T + 228 T^{2} + 796 T^{3} + 326 p T^{4} + 796 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$1 - 280 T^{2} + 40028 T^{4} - 3597992 T^{6} + 303244998 T^{8} - 3597992 p^{2} T^{10} + 40028 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$