# Properties

 Label 16-570e8-1.1-c1e8-0-6 Degree $16$ Conductor $1.114\times 10^{22}$ Sign $1$ Analytic cond. $184168.$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·4-s + 4·5-s + 2·9-s + 16·11-s + 16-s + 8·19-s + 8·20-s + 8·25-s − 8·29-s − 24·31-s + 4·36-s + 32·44-s + 8·45-s + 20·49-s + 64·55-s + 8·59-s + 12·61-s − 2·64-s + 24·71-s + 16·76-s + 28·79-s + 4·80-s + 81-s − 56·89-s + 32·95-s + 32·99-s + 16·100-s + ⋯
 L(s)  = 1 + 4-s + 1.78·5-s + 2/3·9-s + 4.82·11-s + 1/4·16-s + 1.83·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 4.31·31-s + 2/3·36-s + 4.82·44-s + 1.19·45-s + 20/7·49-s + 8.62·55-s + 1.04·59-s + 1.53·61-s − 1/4·64-s + 2.84·71-s + 1.83·76-s + 3.15·79-s + 0.447·80-s + 1/9·81-s − 5.93·89-s + 3.28·95-s + 3.21·99-s + 8/5·100-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$184168.$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$18.19538666$$ $$L(\frac12)$$ $$\approx$$ $$18.19538666$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - T^{2} + T^{4} )^{2}$$
3 $$( 1 - T^{2} + T^{4} )^{2}$$
5 $$1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
19 $$( 1 - 2 T + p T^{2} )^{4}$$
good7 $$( 1 - 5 T^{2} + p^{2} T^{4} )^{4}$$
11 $$( 1 - 2 T + p T^{2} )^{8}$$
13 $$1 + 38 T^{2} + 769 T^{4} + 12806 T^{6} + 183028 T^{8} + 12806 p^{2} T^{10} + 769 p^{4} T^{12} + 38 p^{6} T^{14} + p^{8} T^{16}$$
17 $$( 1 + 10 T^{2} - 189 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
23 $$( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
29 $$( 1 + 4 T + 8 T^{2} - 200 T^{3} - 1241 T^{4} - 200 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}$$
37 $$( 1 - 38 T^{2} + 1923 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
41 $$( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
43 $$1 + 142 T^{2} + 11641 T^{4} + 685150 T^{6} + 31939492 T^{8} + 685150 p^{2} T^{10} + 11641 p^{4} T^{12} + 142 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 + 104 T^{2} + 4558 T^{4} + 191360 T^{6} + 10390339 T^{8} + 191360 p^{2} T^{10} + 4558 p^{4} T^{12} + 104 p^{6} T^{14} + p^{8} T^{16}$$
53 $$1 + 36 T^{2} + 1498 T^{4} - 209520 T^{6} - 11490141 T^{8} - 209520 p^{2} T^{10} + 1498 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 1 - 4 T - 10 T^{2} + 368 T^{3} - 3749 T^{4} + 368 p T^{5} - 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
61 $$( 1 - 6 T - 41 T^{2} + 270 T^{3} + 12 T^{4} + 270 p T^{5} - 41 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 + 254 T^{2} + 39433 T^{4} + 4090670 T^{6} + 319393444 T^{8} + 4090670 p^{2} T^{10} + 39433 p^{4} T^{12} + 254 p^{6} T^{14} + p^{8} T^{16}$$
71 $$( 1 - 12 T - 28 T^{2} - 360 T^{3} + 14319 T^{4} - 360 p T^{5} - 28 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$1 + 2 p T^{2} + 10033 T^{4} + 1250 p T^{6} - 14685116 T^{8} + 1250 p^{3} T^{10} + 10033 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16}$$
79 $$( 1 - 14 T + 85 T^{2} + 658 T^{3} - 9404 T^{4} + 658 p T^{5} + 85 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$( 1 - 152 T^{2} + 11778 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 + 28 T + 416 T^{2} + 5320 T^{3} + 57727 T^{4} + 5320 p T^{5} + 416 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$1 - 76 T^{2} - 662 T^{4} + 940880 T^{6} - 101427821 T^{8} + 940880 p^{2} T^{10} - 662 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$