# Properties

 Label 16-162e8-1.1-c2e8-0-0 Degree $16$ Conductor $4.744\times 10^{17}$ Sign $1$ Analytic cond. $144145.$ Root an. cond. $2.10099$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s − 8·7-s + 64·13-s + 4·16-s + 112·19-s − 64·25-s − 32·28-s − 32·31-s − 152·37-s + 88·43-s + 4·49-s + 256·52-s + 52·61-s − 16·64-s + 40·67-s + 448·73-s + 448·76-s − 104·79-s − 512·91-s − 32·97-s − 256·100-s + 112·103-s − 128·109-s − 32·112-s − 52·121-s − 128·124-s + 127-s + ⋯
 L(s)  = 1 + 4-s − 8/7·7-s + 4.92·13-s + 1/4·16-s + 5.89·19-s − 2.55·25-s − 8/7·28-s − 1.03·31-s − 4.10·37-s + 2.04·43-s + 4/49·49-s + 4.92·52-s + 0.852·61-s − 1/4·64-s + 0.597·67-s + 6.13·73-s + 5.89·76-s − 1.31·79-s − 5.62·91-s − 0.329·97-s − 2.55·100-s + 1.08·103-s − 1.17·109-s − 2/7·112-s − 0.429·121-s − 1.03·124-s + 0.00787·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{32}$$ Sign: $1$ Analytic conductor: $$144145.$$ Root analytic conductor: $$2.10099$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$9.955567540$$ $$L(\frac12)$$ $$\approx$$ $$9.955567540$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
3 $$1$$
good5 $$1 + 64 T^{2} + 413 p T^{4} + 49984 T^{6} + 1174336 T^{8} + 49984 p^{4} T^{10} + 413 p^{9} T^{12} + 64 p^{12} T^{14} + p^{16} T^{16}$$
7 $$( 1 + 4 T + 22 T^{2} - 416 T^{3} - 3149 T^{4} - 416 p^{2} T^{5} + 22 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
11 $$( 1 + 26 T^{2} - 13965 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 - 32 T + 457 T^{2} - 7328 T^{3} + 119872 T^{4} - 7328 p^{2} T^{5} + 457 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
17 $$( 1 - 256 T^{2} + 31551 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
19 $$( 1 - 28 T + 810 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
23 $$( 1 + 842 T^{2} + 429123 T^{4} + 842 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
29 $$1 + 1312 T^{2} - 68879 T^{4} + 492867232 T^{6} + 1487178071104 T^{8} + 492867232 p^{4} T^{10} - 68879 p^{8} T^{12} + 1312 p^{12} T^{14} + p^{16} T^{16}$$
31 $$( 1 + 8 T - 897 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
37 $$( 1 + 38 T + 1371 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
41 $$1 + 4960 T^{2} + 12861886 T^{4} + 30197432320 T^{6} + 60317717755075 T^{8} + 30197432320 p^{4} T^{10} + 12861886 p^{8} T^{12} + 4960 p^{12} T^{14} + p^{16} T^{16}$$
43 $$( 1 - 44 T - 1274 T^{2} + 21472 T^{3} + 3305635 T^{4} + 21472 p^{2} T^{5} - 1274 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
47 $$( 1 + 4130 T^{2} + 12177219 T^{4} + 4130 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 3424 T^{2} + 8198754 T^{4} - 3424 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$1 + 1828 T^{2} - 16689686 T^{4} - 7683910256 T^{6} + 213930160568755 T^{8} - 7683910256 p^{4} T^{10} - 16689686 p^{8} T^{12} + 1828 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 - 13 T - 3552 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
67 $$( 1 - 20 T - 7706 T^{2} + 17440 T^{3} + 43760515 T^{4} + 17440 p^{2} T^{5} - 7706 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
71 $$( 1 - 9652 T^{2} + 49230438 T^{4} - 9652 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 112 T + 13551 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
79 $$( 1 + 52 T - 5162 T^{2} - 240032 T^{3} + 6048211 T^{4} - 240032 p^{2} T^{5} - 5162 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
83 $$1 + 16612 T^{2} + 139485994 T^{4} + 690326743696 T^{6} + 3275935507518835 T^{8} + 690326743696 p^{4} T^{10} + 139485994 p^{8} T^{12} + 16612 p^{12} T^{14} + p^{16} T^{16}$$
89 $$( 1 - 7456 T^{2} + 104845119 T^{4} - 7456 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
97 $$( 1 + 16 T - 11714 T^{2} - 109568 T^{3} + 52342915 T^{4} - 109568 p^{2} T^{5} - 11714 p^{4} T^{6} + 16 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}$$