# Properties

 Label 16-300e8-1.1-c2e8-0-4 Degree $16$ Conductor $6.561\times 10^{19}$ Sign $1$ Analytic cond. $1.99365\times 10^{7}$ Root an. cond. $2.85909$ 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 − 12·9-s + 16-s + 184·29-s − 60·36-s − 256·41-s + 208·49-s + 304·61-s − 35·64-s + 90·81-s − 560·89-s + 296·101-s + 608·109-s + 920·116-s + 272·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s − 1.28e3·164-s + 167-s − 1.18e3·169-s + ⋯
 L(s)  = 1 + 5/4·4-s − 4/3·9-s + 1/16·16-s + 6.34·29-s − 5/3·36-s − 6.24·41-s + 4.24·49-s + 4.98·61-s − 0.546·64-s + 10/9·81-s − 6.29·89-s + 2.93·101-s + 5.57·109-s + 7.93·116-s + 2.24·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.0833·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s − 7.80·164-s + 0.00598·167-s − 7.00·169-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$16$$ Conductor: $$2^{16} \cdot 3^{8} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$1.99365\times 10^{7}$$ Root analytic conductor: $$2.85909$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{300} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$7.127897911$$ $$L(\frac12)$$ $$\approx$$ $$7.127897911$$ $$L(2)$$ 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 + p T^{2} )^{4}$$
5 $$1$$
good7 $$( 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}$$