# Properties

 Label 8-384e4-1.1-c3e4-0-0 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $263505.$ Root an. cond. $4.75990$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 50·9-s + 352·23-s − 292·25-s − 1.53e3·47-s + 1.16e3·49-s − 2.72e3·71-s + 1.68e3·73-s + 1.77e3·81-s − 4.24e3·97-s + 1.09e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.91e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
 L(s)  = 1 + 1.85·9-s + 3.19·23-s − 2.33·25-s − 4.76·47-s + 3.39·49-s − 4.54·71-s + 2.70·73-s + 2.42·81-s − 4.44·97-s + 0.820·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.23·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$263505.$$ Root analytic conductor: $$4.75990$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.01102213192$$ $$L(\frac12)$$ $$\approx$$ $$0.01102213192$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_2^2$ $$1 - 50 T^{2} + p^{6} T^{4}$$
good5$C_2^2$ $$( 1 + 146 T^{2} + p^{6} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 582 T^{2} + p^{6} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 546 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 2458 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 9410 T^{2} + p^{6} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 13614 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2$ $$( 1 - 88 T + p^{3} T^{2} )^{4}$$
29$C_2^2$ $$( 1 - 16222 T^{2} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 13718 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 8918 T^{2} + p^{6} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 101774 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2^2$ $$( 1 + 103998 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2$ $$( 1 + 384 T + p^{3} T^{2} )^{4}$$
53$C_2^2$ $$( 1 + 87154 T^{2} + p^{6} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 13858 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 398266 T^{2} + p^{6} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 598926 T^{2} + p^{6} T^{4} )^{2}$$
71$C_2$ $$( 1 + 680 T + p^{3} T^{2} )^{4}$$
73$C_2$ $$( 1 - 422 T + p^{3} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 431862 T^{2} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 1108978 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 490994 T^{2} + p^{6} T^{4} )^{2}$$
97$C_2$ $$( 1 + 1062 T + p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$