# Properties

 Label 16-63e8-1.1-c5e8-0-0 Degree $16$ Conductor $2.482\times 10^{14}$ Sign $1$ Analytic cond. $1.08644\times 10^{8}$ Root an. cond. $3.17870$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 128·4-s − 112·7-s + 8.16e3·16-s − 7.81e3·25-s − 1.43e4·28-s − 3.43e4·37-s + 3.24e4·43-s − 4.67e4·49-s + 3.87e5·64-s + 1.52e5·67-s − 9.45e4·79-s − 1.00e6·100-s − 6.41e5·109-s − 9.13e5·112-s + 1.12e6·121-s + 127-s + 131-s + 137-s + 139-s − 4.39e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.61e6·169-s + 4.14e6·172-s + ⋯
 L(s)  = 1 + 4·4-s − 0.863·7-s + 7.96·16-s − 2.50·25-s − 3.45·28-s − 4.11·37-s + 2.67·43-s − 2.78·49-s + 11.8·64-s + 4.15·67-s − 1.70·79-s − 10.0·100-s − 5.16·109-s − 6.88·112-s + 7.01·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 16.4·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4.35·169-s + 10.6·172-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$3^{16} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$1.08644\times 10^{8}$$ Root analytic conductor: $$3.17870$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$10.82514849$$ $$L(\frac12)$$ $$\approx$$ $$10.82514849$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$( 1 + 8 p T + 4014 p T^{2} + 8 p^{6} T^{3} + p^{10} T^{4} )^{2}$$
good2 $$( 1 - p^{6} T^{2} + 129 p^{4} T^{4} - p^{16} T^{6} + p^{20} T^{8} )^{2}$$
5 $$( 1 + 3908 T^{2} + 23248566 T^{4} + 3908 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
11 $$( 1 - 564976 T^{2} + 131629015794 T^{4} - 564976 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
13 $$( 1 - 809044 T^{2} + 351572118390 T^{4} - 809044 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
17 $$( 1 + 1883444 T^{2} + 1329835154982 T^{4} + 1883444 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
19 $$( 1 - 190372 p T^{2} + 10651698082806 T^{4} - 190372 p^{11} T^{6} + p^{20} T^{8} )^{2}$$
23 $$( 1 - 4853968 T^{2} + 70775754302754 T^{4} - 4853968 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
29 $$( 1 - 61078432 T^{2} + 1759165582775058 T^{4} - 61078432 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
31 $$( 1 - 23985916 T^{2} + 1685938631161734 T^{4} - 23985916 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
37 $$( 1 + 8576 T + 87352506 T^{2} + 8576 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
41 $$( 1 + 13154708 T^{2} - 843745561706682 T^{4} + 13154708 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
43 $$( 1 - 8104 T + 215741802 T^{2} - 8104 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
47 $$( 1 + 102260732 T^{2} + 1855613853315738 p T^{4} + 102260732 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
53 $$( 1 - 334196128 T^{2} + 373722291304309746 T^{4} - 334196128 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
59 $$( 1 + 1446882860 T^{2} + 1156390163071869654 T^{4} + 1446882860 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
61 $$( 1 - 2484649012 T^{2} + 2966879913883883766 T^{4} - 2484649012 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
67 $$( 1 - 38152 T + 1428083382 T^{2} - 38152 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
71 $$( 1 - 1243284688 T^{2} - 1234162537544764062 T^{4} - 1243284688 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
73 $$( 1 - 4781796484 T^{2} + 14254537270266844710 T^{4} - 4781796484 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
79 $$( 1 + 23648 T + 5558769246 T^{2} + 23648 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
83 $$( 1 + 10148193356 T^{2} + 54914978401780671414 T^{4} + 10148193356 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
89 $$( 1 + 9743291348 T^{2} + 65866481293586129286 T^{4} + 9743291348 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
97 $$( 1 - 23990990692 T^{2} +$$$$26\!\cdots\!46$$$$T^{4} - 23990990692 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$