# Properties

 Label 8-90e8-1.1-c1e4-0-2 Degree $8$ Conductor $4.305\times 10^{15}$ Sign $1$ Analytic cond. $1.75004\times 10^{7}$ Root an. cond. $8.04231$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·19-s − 24·29-s − 4·31-s + 24·41-s + 20·49-s − 24·59-s − 16·61-s + 24·71-s + 16·79-s + 24·101-s + 4·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 0.917·19-s − 4.45·29-s − 0.718·31-s + 3.74·41-s + 20/7·49-s − 3.12·59-s − 2.04·61-s + 2.84·71-s + 1.80·79-s + 2.38·101-s + 0.383·109-s − 3.45·121-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 + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 3^{16} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$1.75004\times 10^{7}$$ Root analytic conductor: $$8.04231$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{8100} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.457607946$$ $$L(\frac12)$$ $$\approx$$ $$1.457607946$$ $$L(\frac{3}{2})$$ 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 $$1$$
5 $$1$$
good7$D_4\times C_2$ $$1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 44 T^{2} + 954 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 34 T^{2} + p^{2} T^{4} )^{2}$$
29$D_{4}$ $$( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 92 T^{2} + 4746 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 116 T^{2} + 6762 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
67$D_4\times C_2$ $$1 - 20 T^{2} + 2166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 188 T^{2} + 16794 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 + 151 T^{2} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 380 T^{2} + 54906 T^{4} - 380 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$