# Properties

 Label 16-1840e8-1.1-c1e8-0-1 Degree $16$ Conductor $1.314\times 10^{26}$ Sign $1$ Analytic cond. $2.17149\times 10^{9}$ Root an. cond. $3.83307$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·5-s + 8·9-s − 4·11-s − 8·19-s + 10·25-s − 8·29-s − 16·41-s − 48·45-s + 28·49-s + 24·55-s − 16·61-s + 48·71-s + 48·79-s + 30·81-s + 16·89-s + 48·95-s − 32·99-s − 24·101-s − 8·109-s − 44·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯
 L(s)  = 1 − 2.68·5-s + 8/3·9-s − 1.20·11-s − 1.83·19-s + 2·25-s − 1.48·29-s − 2.49·41-s − 7.15·45-s + 4·49-s + 3.23·55-s − 2.04·61-s + 5.69·71-s + 5.40·79-s + 10/3·81-s + 1.69·89-s + 4.92·95-s − 3.21·99-s − 2.38·101-s − 0.766·109-s − 4·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{32} \cdot 5^{8} \cdot 23^{8}$$ Sign: $1$ Analytic conductor: $$2.17149\times 10^{9}$$ Root analytic conductor: $$3.83307$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1840} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{32} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2785569766$$ $$L(\frac12)$$ $$\approx$$ $$0.2785569766$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + 6 T + 26 T^{2} + 82 T^{3} + 206 T^{4} + 82 p T^{5} + 26 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
23 $$( 1 + T^{2} )^{4}$$
good3 $$1 - 8 T^{2} + 34 T^{4} - 32 p T^{6} + 259 T^{8} - 32 p^{3} T^{10} + 34 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16}$$
7 $$1 - 4 p T^{2} + 348 T^{4} - 2708 T^{6} + 18230 T^{8} - 2708 p^{2} T^{10} + 348 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}$$
11 $$( 1 + 2 T + 28 T^{2} + 2 p T^{3} + 346 T^{4} + 2 p^{2} T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
13 $$1 - 40 T^{2} + 914 T^{4} - 15968 T^{6} + 228595 T^{8} - 15968 p^{2} T^{10} + 914 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16}$$
17 $$1 - 56 T^{2} + 1544 T^{4} - 36168 T^{6} + 720894 T^{8} - 36168 p^{2} T^{10} + 1544 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16}$$
19 $$( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 200 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
29 $$( 1 + 4 T + 94 T^{2} + 344 T^{3} + 3775 T^{4} + 344 p T^{5} + 94 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 50 T^{2} + 256 T^{3} + 1011 T^{4} + 256 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
37 $$1 - 4 p T^{2} + 11228 T^{4} - 572828 T^{6} + 23118550 T^{8} - 572828 p^{2} T^{10} + 11228 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16}$$
41 $$( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 616 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
43 $$1 - 92 T^{2} + 8096 T^{4} - 11164 p T^{6} + 22994446 T^{8} - 11164 p^{3} T^{10} + 8096 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 - 96 T^{2} + 11626 T^{4} - 665136 T^{6} + 41700939 T^{8} - 665136 p^{2} T^{10} + 11626 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16}$$
53 $$1 - 200 T^{2} + 23516 T^{4} - 1842744 T^{6} + 111792678 T^{8} - 1842744 p^{2} T^{10} + 23516 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 16 p T^{5} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 316 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 - 516 T^{2} + 117772 T^{4} - 15527980 T^{6} + 1293477494 T^{8} - 15527980 p^{2} T^{10} + 117772 p^{4} T^{12} - 516 p^{6} T^{14} + p^{8} T^{16}$$
71 $$( 1 - 24 T + 350 T^{2} - 3544 T^{3} + 32183 T^{4} - 3544 p T^{5} + 350 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$1 - 216 T^{2} + 38578 T^{4} - 4059072 T^{6} + 362257971 T^{8} - 4059072 p^{2} T^{10} + 38578 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 6100 p T^{5} + 494 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$1 - 88 T^{2} + 20552 T^{4} - 1491272 T^{6} + 193546270 T^{8} - 1491272 p^{2} T^{10} + 20552 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16}$$
89 $$( 1 - 8 T + 270 T^{2} - 1764 T^{3} + 34598 T^{4} - 1764 p T^{5} + 270 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$1 - 316 T^{2} + 65324 T^{4} - 9550388 T^{6} + 1045123990 T^{8} - 9550388 p^{2} T^{10} + 65324 p^{4} T^{12} - 316 p^{6} T^{14} + p^{8} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$