Properties

 Label 8-4608e4-1.1-c1e4-0-34 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 $4$

Origins of factors

Dirichlet series

 L(s)  = 1 − 8·5-s + 32·25-s − 8·29-s − 40·53-s − 8·97-s − 56·101-s − 20·121-s − 104·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 3.57·5-s + 32/5·25-s − 1.48·29-s − 5.49·53-s − 0.812·97-s − 5.57·101-s − 1.81·121-s − 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-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: $$4$$ Selberg data: $$(8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2^3$ $$1 + 2 T^{4} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 10 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 + p T^{2} )^{4}$$
17$C_2$ $$( 1 + p T^{2} )^{4}$$
19$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2$ $$( 1 + p T^{2} )^{4}$$
29$C_2^2$ $$( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^3$ $$1 - 478 T^{4} + p^{4} T^{8}$$
37$C_2$ $$( 1 + p T^{2} )^{4}$$
41$C_2$ $$( 1 + p T^{2} )^{4}$$
43$C_2$ $$( 1 + p T^{2} )^{4}$$
47$C_2$ $$( 1 + p T^{2} )^{4}$$
53$C_2^2$ $$( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + p T^{2} )^{4}$$
67$C_2$ $$( 1 + p T^{2} )^{4}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^3$ $$1 - 9118 T^{4} + p^{4} T^{8}$$
83$C_2^2$ $$( 1 - 134 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 + p T^{2} )^{4}$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$