# Properties

 Label 16-1620e8-1.1-c2e8-0-1 Degree $16$ Conductor $4.744\times 10^{25}$ Sign $1$ Analytic cond. $1.44145\times 10^{13}$ Root an. cond. $6.64392$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·5-s − 12·17-s + 12·19-s − 30·23-s − 16·31-s + 48·47-s − 137·49-s + 384·53-s − 38·61-s + 6·79-s − 288·83-s − 36·85-s + 36·95-s − 36·107-s − 452·109-s − 564·113-s − 90·115-s − 129·121-s − 177·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + 157-s + ⋯
 L(s)  = 1 + 3/5·5-s − 0.705·17-s + 0.631·19-s − 1.30·23-s − 0.516·31-s + 1.02·47-s − 2.79·49-s + 7.24·53-s − 0.622·61-s + 6/79·79-s − 3.46·83-s − 0.423·85-s + 0.378·95-s − 0.336·107-s − 4.14·109-s − 4.99·113-s − 0.782·115-s − 1.06·121-s − 1.41·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.309·155-s + 0.00636·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 3^{32} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$1.44145\times 10^{13}$$ Root analytic conductor: $$6.64392$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1620} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.09743664129$$ $$L(\frac12)$$ $$\approx$$ $$0.09743664129$$ $$L(2)$$ 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$$
5 $$1 - 3 T + 9 T^{2} + 6 p^{2} T^{3} - 34 p^{2} T^{4} + 6 p^{4} T^{5} + 9 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8}$$
good7 $$1 + 137 T^{2} + 9745 T^{4} + 578414 T^{6} + 30603406 T^{8} + 578414 p^{4} T^{10} + 9745 p^{8} T^{12} + 137 p^{12} T^{14} + p^{16} T^{16}$$
11 $$1 + 129 T^{2} + 6241 T^{4} - 2435778 T^{6} - 349839762 T^{8} - 2435778 p^{4} T^{10} + 6241 p^{8} T^{12} + 129 p^{12} T^{14} + p^{16} T^{16}$$
13 $$1 + 136 T^{2} + 1882 p T^{4} - 8580512 T^{6} - 1308354077 T^{8} - 8580512 p^{4} T^{10} + 1882 p^{9} T^{12} + 136 p^{12} T^{14} + p^{16} T^{16}$$
17 $$( 1 + 3 T + 528 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
19 $$( 1 - 3 T + 254 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
23 $$( 1 + 15 T - 837 T^{2} + 60 T^{3} + 728978 T^{4} + 60 p^{2} T^{5} - 837 p^{4} T^{6} + 15 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
29 $$( 1 + 98 T^{2} - 697677 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
31 $$( 1 + 8 T + 7 T^{2} - 14920 T^{3} - 981776 T^{4} - 14920 p^{2} T^{5} + 7 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
37 $$( 1 - 1522 T^{2} + p^{4} T^{4} )^{4}$$
41 $$( 1 + 3186 T^{2} + 7324835 T^{4} + 3186 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
43 $$1 + 776 T^{2} - 5114414 T^{4} - 869905312 T^{6} + 18932481039523 T^{8} - 869905312 p^{4} T^{10} - 5114414 p^{8} T^{12} + 776 p^{12} T^{14} + p^{16} T^{16}$$
47 $$( 1 - 24 T - 3150 T^{2} + 16608 T^{3} + 7731011 T^{4} + 16608 p^{2} T^{5} - 3150 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 96 T + 6041 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
59 $$1 + 7044 T^{2} + 20683306 T^{4} + 33106151952 T^{6} + 89143933045683 T^{8} + 33106151952 p^{4} T^{10} + 20683306 p^{8} T^{12} + 7044 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 + 19 T - 6701 T^{2} - 7220 T^{3} + 34682722 T^{4} - 7220 p^{2} T^{5} - 6701 p^{4} T^{6} + 19 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
67 $$1 + 6584 T^{2} + 23998450 T^{4} - 137945571424 T^{6} - 905153797681181 T^{8} - 137945571424 p^{4} T^{10} + 23998450 p^{8} T^{12} + 6584 p^{12} T^{14} + p^{16} T^{16}$$
71 $$( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
79 $$( 1 - 3 T - 8243 T^{2} + 12690 T^{3} + 29089254 T^{4} + 12690 p^{2} T^{5} - 8243 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
83 $$( 1 + 144 T + 6999 T^{2} - 5904 T^{3} - 1603456 T^{4} - 5904 p^{2} T^{5} + 6999 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
89 $$( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
97 $$1 - 1879 T^{2} - 50655359 T^{4} + 230877543998 T^{6} - 5213876107576082 T^{8} + 230877543998 p^{4} T^{10} - 50655359 p^{8} T^{12} - 1879 p^{12} T^{14} + p^{16} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$