# Properties

 Label 16-416e8-1.1-c1e8-0-0 Degree $16$ Conductor $8.969\times 10^{20}$ Sign $1$ Analytic cond. $14823.9$ Root an. cond. $1.82257$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·9-s − 8·13-s − 12·17-s + 24·25-s − 12·29-s + 12·37-s − 36·41-s − 6·49-s + 48·53-s − 12·61-s + 7·81-s − 36·89-s + 12·97-s + 36·101-s + 36·113-s − 16·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 + 2/3·9-s − 2.21·13-s − 2.91·17-s + 24/5·25-s − 2.22·29-s + 1.97·37-s − 5.62·41-s − 6/7·49-s + 6.59·53-s − 1.53·61-s + 7/9·81-s − 3.81·89-s + 1.21·97-s + 3.58·101-s + 3.38·113-s − 1.47·117-s − 1.27·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{40} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$14823.9$$ Root analytic conductor: $$1.82257$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{40} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6355750485$$ $$L(\frac12)$$ $$\approx$$ $$0.6355750485$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$( 1 + 2 T + p T^{2} )^{4}$$
good3 $$1 - 2 T^{2} - p T^{4} + 22 T^{6} - 68 T^{8} + 22 p^{2} T^{10} - p^{5} T^{12} - 2 p^{6} T^{14} + p^{8} T^{16}$$
5 $$( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4}$$
7 $$1 + 6 T^{2} + 37 T^{4} - 594 T^{6} - 4164 T^{8} - 594 p^{2} T^{10} + 37 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16}$$
11 $$1 + 14 T^{2} + 13 T^{4} - 826 T^{6} - 4868 T^{8} - 826 p^{2} T^{10} + 13 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16}$$
17 $$( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
19 $$1 + 54 T^{2} + 1573 T^{4} + 33534 T^{6} + 620652 T^{8} + 33534 p^{2} T^{10} + 1573 p^{4} T^{12} + 54 p^{6} T^{14} + p^{8} T^{16}$$
23 $$1 - 10 T^{2} - 875 T^{4} + 830 T^{6} + 617884 T^{8} + 830 p^{2} T^{10} - 875 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16}$$
29 $$( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4}$$
31 $$( 1 - 36 T^{2} + 518 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
37 $$( 1 - 2 T + p T^{2} )^{4}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
41 $$( 1 + 18 T + 213 T^{2} + 1890 T^{3} + 13772 T^{4} + 1890 p T^{5} + 213 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
43 $$1 - 82 T^{2} + 2317 T^{4} - 58138 T^{6} + 3570172 T^{8} - 58138 p^{2} T^{10} + 2317 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16}$$
47 $$( 1 + 4 T^{2} + 2694 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
53 $$( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4}$$
59 $$1 - 10 T^{2} - 5915 T^{4} + 9470 T^{6} + 23714764 T^{8} + 9470 p^{2} T^{10} - 5915 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16}$$
61 $$( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4}$$
67 $$1 + 246 T^{2} + 36517 T^{4} + 3695166 T^{6} + 284144556 T^{8} + 3695166 p^{2} T^{10} + 36517 p^{4} T^{12} + 246 p^{6} T^{14} + p^{8} T^{16}$$
71 $$1 + 14 T^{2} - 1187 T^{4} - 121786 T^{6} - 24487028 T^{8} - 121786 p^{2} T^{10} - 1187 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16}$$
73 $$( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4}$$
79 $$( 1 + 196 T^{2} + 20358 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
83 $$( 1 - 212 T^{2} + 23286 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 + 18 T + 297 T^{2} + 3402 T^{3} + 37412 T^{4} + 3402 p T^{5} + 297 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$( 1 - 6 T + 65 T^{2} - 318 T^{3} - 5436 T^{4} - 318 p T^{5} + 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$