# Properties

 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Motivic weight $2$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 8·7-s + 8·9-s − 24·19-s + 32·21-s + 80·25-s − 4·27-s − 56·31-s + 32·37-s + 136·43-s − 128·49-s − 96·57-s − 160·61-s + 64·63-s − 280·67-s − 80·73-s + 320·75-s − 408·79-s − 30·81-s − 224·93-s + 96·97-s + 488·103-s + 160·109-s + 128·111-s + 480·121-s + 127-s + 544·129-s + ⋯
 L(s)  = 1 + 4/3·3-s + 8/7·7-s + 8/9·9-s − 1.26·19-s + 1.52·21-s + 16/5·25-s − 0.148·27-s − 1.80·31-s + 0.864·37-s + 3.16·43-s − 2.61·49-s − 1.68·57-s − 2.62·61-s + 1.01·63-s − 4.17·67-s − 1.09·73-s + 4.26·75-s − 5.16·79-s − 0.370·81-s − 2.40·93-s + 0.989·97-s + 4.73·103-s + 1.46·109-s + 1.15·111-s + 3.96·121-s + 0.00787·127-s + 4.21·129-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Motivic weight: $$2$$ Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.468725$$ $$L(\frac12)$$ $$\approx$$ $$0.468725$$ $$L(2)$$ 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 - 4 T + 8 T^{2} + 4 T^{3} - 22 p T^{4} + 4 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8}$$
good5 $$1 - 16 p T^{2} + 3004 T^{4} - 18672 p T^{6} + 2634054 T^{8} - 18672 p^{5} T^{10} + 3004 p^{8} T^{12} - 16 p^{13} T^{14} + p^{16} T^{16}$$
7 $$( 1 - 4 T + 88 T^{2} - 220 T^{3} + 3646 T^{4} - 220 p^{2} T^{5} + 88 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
11 $$1 - 480 T^{2} + 125212 T^{4} - 22531104 T^{6} + 3078264198 T^{8} - 22531104 p^{4} T^{10} + 125212 p^{8} T^{12} - 480 p^{12} T^{14} + p^{16} T^{16}$$
13 $$( 1 + 316 T^{2} + 256 T^{3} + 57510 T^{4} + 256 p^{2} T^{5} + 316 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
17 $$1 - 1064 T^{2} + 654556 T^{4} - 282654744 T^{6} + 92440824774 T^{8} - 282654744 p^{4} T^{10} + 654556 p^{8} T^{12} - 1064 p^{12} T^{14} + p^{16} T^{16}$$
19 $$( 1 + 12 T + 664 T^{2} - 4588 T^{3} + 120414 T^{4} - 4588 p^{2} T^{5} + 664 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
23 $$1 - 1896 T^{2} + 2093020 T^{4} - 1543350744 T^{6} + 914302437702 T^{8} - 1543350744 p^{4} T^{10} + 2093020 p^{8} T^{12} - 1896 p^{12} T^{14} + p^{16} T^{16}$$
29 $$1 - 3024 T^{2} + 4188220 T^{4} - 3460574000 T^{6} + 2571955018182 T^{8} - 3460574000 p^{4} T^{10} + 4188220 p^{8} T^{12} - 3024 p^{12} T^{14} + p^{16} T^{16}$$
31 $$( 1 + 28 T + 3016 T^{2} + 63204 T^{3} + 3999966 T^{4} + 63204 p^{2} T^{5} + 3016 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
37 $$( 1 - 16 T + 1948 T^{2} + 25232 T^{3} + 1096870 T^{4} + 25232 p^{2} T^{5} + 1948 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
41 $$1 - 2472 T^{2} + 3015004 T^{4} - 2001918872 T^{6} + 1122734027334 T^{8} - 2001918872 p^{4} T^{10} + 3015004 p^{8} T^{12} - 2472 p^{12} T^{14} + p^{16} T^{16}$$
43 $$( 1 - 68 T + 6808 T^{2} - 284700 T^{3} + 17420958 T^{4} - 284700 p^{2} T^{5} + 6808 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
47 $$1 - 12296 T^{2} + 72283420 T^{4} - 268133584440 T^{6} + 697058148522822 T^{8} - 268133584440 p^{4} T^{10} + 72283420 p^{8} T^{12} - 12296 p^{12} T^{14} + p^{16} T^{16}$$
53 $$1 + 304 T^{2} + 25898428 T^{4} + 6393839056 T^{6} + 288293861465158 T^{8} + 6393839056 p^{4} T^{10} + 25898428 p^{8} T^{12} + 304 p^{12} T^{14} + p^{16} T^{16}$$
59 $$1 - 19008 T^{2} + 181109212 T^{4} - 1092163345344 T^{6} + 4537129454477574 T^{8} - 1092163345344 p^{4} T^{10} + 181109212 p^{8} T^{12} - 19008 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 + 80 T + 12700 T^{2} + 630832 T^{3} + 62237030 T^{4} + 630832 p^{2} T^{5} + 12700 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
67 $$( 1 + 140 T + 18856 T^{2} + 1550068 T^{3} + 122202110 T^{4} + 1550068 p^{2} T^{5} + 18856 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
71 $$1 - 17640 T^{2} + 102948316 T^{4} - 54971476568 T^{6} - 1448379553007034 T^{8} - 54971476568 p^{4} T^{10} + 102948316 p^{8} T^{12} - 17640 p^{12} T^{14} + p^{16} T^{16}$$
73 $$( 1 + 40 T + 15772 T^{2} + 446872 T^{3} + 110402182 T^{4} + 446872 p^{2} T^{5} + 15772 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
79 $$( 1 + 204 T + 32488 T^{2} + 3143092 T^{3} + 288852126 T^{4} + 3143092 p^{2} T^{5} + 32488 p^{4} T^{6} + 204 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
83 $$1 - 41696 T^{2} + 833769628 T^{4} - 10294236337952 T^{6} + 85511640874023430 T^{8} - 10294236337952 p^{4} T^{10} + 833769628 p^{8} T^{12} - 41696 p^{12} T^{14} + p^{16} T^{16}$$
89 $$1 - 17736 T^{2} + 318384412 T^{4} - 3335315895288 T^{6} + 31854516109881798 T^{8} - 3335315895288 p^{4} T^{10} + 318384412 p^{8} T^{12} - 17736 p^{12} T^{14} + p^{16} T^{16}$$
97 $$( 1 - 48 T + 29500 T^{2} - 1080784 T^{3} + 386488710 T^{4} - 1080784 p^{2} T^{5} + 29500 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$