# Properties

 Label 8-4608e4-1.1-c1e4-0-8 Degree $8$ Conductor $4.509\times 10^{14}$ Sign $1$ Analytic cond. $1.83298\times 10^{6}$ Root an. cond. $6.06589$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·11-s − 8·17-s − 24·41-s + 16·43-s + 16·59-s + 32·67-s + 16·73-s + 24·83-s − 40·89-s + 24·97-s − 16·107-s − 24·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s − 64·187-s + ⋯
 L(s)  = 1 + 2.41·11-s − 1.94·17-s − 3.74·41-s + 2.43·43-s + 2.08·59-s + 3.90·67-s + 1.87·73-s + 2.63·83-s − 4.23·89-s + 2.43·97-s − 1.54·107-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 4.68·187-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{36} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$1.83298\times 10^{6}$$ Root analytic conductor: $$6.06589$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{4608} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.123099769$$ $$L(\frac12)$$ $$\approx$$ $$2.123099769$$ $$L(\frac{3}{2})$$ 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 $$1$$
good5$D_4\times C_2$ $$1 + 18 T^{4} + p^{4} T^{8}$$
7$D_4\times C_2$ $$1 - 30 T^{4} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2 \wr C_2$ $$1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
17$D_{4}$ $$( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 30 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 + 12 T^{2} + 582 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 1650 T^{4} + p^{4} T^{8}$$
31$C_2^2 \wr C_2$ $$1 + 96 T^{2} + 4098 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2^2 \wr C_2$ $$1 + 92 T^{2} + 4342 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 + 108 T^{2} + 6822 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2 \wr C_2$ $$1 + 64 T^{2} + 1234 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2 \wr C_2$ $$1 + 188 T^{2} + 15766 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 + 172 T^{2} + 15430 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
79$C_2^2 \wr C_2$ $$1 + 288 T^{2} + 33090 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
97$D_{4}$ $$( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$