# Properties

 Label 8-384e4-1.1-c1e4-0-4 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $88.3961$ Root an. cond. $1.75107$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 2·9-s + 12·11-s − 8·23-s + 8·25-s − 6·27-s − 24·33-s − 16·37-s − 32·47-s + 16·49-s + 4·59-s + 16·61-s + 16·69-s − 8·71-s − 8·73-s − 16·75-s + 11·81-s + 20·83-s − 16·97-s + 24·99-s − 12·107-s − 16·109-s + 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 1.15·3-s + 2/3·9-s + 3.61·11-s − 1.66·23-s + 8/5·25-s − 1.15·27-s − 4.17·33-s − 2.63·37-s − 4.66·47-s + 16/7·49-s + 0.520·59-s + 2.04·61-s + 1.92·69-s − 0.949·71-s − 0.936·73-s − 1.84·75-s + 11/9·81-s + 2.19·83-s − 1.62·97-s + 2.41·99-s − 1.16·107-s − 1.53·109-s + 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$88.3961$$ Root analytic conductor: $$1.75107$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.368055916$$ $$L(\frac12)$$ $$\approx$$ $$1.368055916$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
good5$D_4\times C_2$ $$1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
7$D_4\times C_2$ $$1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}$$
11$C_4$ $$( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 6 T^{2} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
53$D_4\times C_2$ $$1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
79$D_4\times C_2$ $$1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 162 T^{2} + p^{2} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$