# Properties

 Label 16-180e8-1.1-c2e8-0-2 Degree $16$ Conductor $1.102\times 10^{18}$ Sign $1$ Analytic cond. $334857.$ Root an. cond. $2.21464$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5·4-s − 4·5-s + 16-s + 20·20-s + 24·25-s + 184·29-s + 256·41-s − 208·49-s + 304·61-s + 35·64-s − 4·80-s − 560·89-s − 120·100-s − 296·101-s − 608·109-s − 920·116-s + 272·121-s − 204·125-s + 127-s + 131-s + 137-s + 139-s − 736·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
 L(s)  = 1 − 5/4·4-s − 4/5·5-s + 1/16·16-s + 20-s + 0.959·25-s + 6.34·29-s + 6.24·41-s − 4.24·49-s + 4.98·61-s + 0.546·64-s − 0.0499·80-s − 6.29·89-s − 6/5·100-s − 2.93·101-s − 5.57·109-s − 7.93·116-s + 2.24·121-s − 1.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.07·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 3^{16} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$334857.$$ Root analytic conductor: $$2.21464$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{180} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.258893563$$ $$L(\frac12)$$ $$\approx$$ $$3.258893563$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 5 T^{2} + 3 p^{3} T^{4} + 5 p^{4} T^{6} + p^{8} T^{8}$$
3 $$1$$
5 $$( 1 + 2 T - 6 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
good7 $$( 1 + 104 T^{2} + 5454 T^{4} + 104 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
11 $$( 1 - 136 T^{2} + 28206 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
13 $$( 1 - 592 T^{2} + 144510 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
17 $$( 1 - 112 T^{2} + 118878 T^{4} - 112 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
19 $$( 1 - 436 T^{2} + 275334 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
23 $$( 1 + 76 p T^{2} + 1290726 T^{4} + 76 p^{5} T^{6} + p^{8} T^{8} )^{2}$$
29 $$( 1 - 46 T + 1698 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
31 $$( 1 - 1540 T^{2} + 1506054 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
37 $$( 1 - 4720 T^{2} + 9299454 T^{4} - 4720 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
41 $$( 1 - 64 T + 2334 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
43 $$( 1 + 4868 T^{2} + 12528486 T^{4} + 4868 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
47 $$( 1 + 5540 T^{2} + 15331014 T^{4} + 5540 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
53 $$( 1 - 10480 T^{2} + 43220094 T^{4} - 10480 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
59 $$( 1 + 2744 T^{2} + 25695534 T^{4} + 2744 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
61 $$( 1 - 38 T + p^{2} T^{2} )^{8}$$
67 $$( 1 + 8564 T^{2} + 44158854 T^{4} + 8564 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
71 $$( 1 - 3028 T^{2} - 19410330 T^{4} - 3028 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
73 $$( 1 - 17140 T^{2} + 129420582 T^{4} - 17140 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
79 $$( 1 - 22660 T^{2} + 205335174 T^{4} - 22660 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
83 $$( 1 + 23492 T^{2} + 230783910 T^{4} + 23492 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
89 $$( 1 + 140 T + 19830 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{4}$$
97 $$( 1 + 8252 T^{2} + 163205766 T^{4} + 8252 p^{4} T^{6} + p^{8} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$