Properties

 Degree $8$ Conductor $21743271936$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 6·9-s − 4·25-s − 20·49-s + 56·73-s + 27·81-s + 8·97-s + 20·121-s + 127-s + 131-s + 137-s + 139-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 + 223-s + 24·225-s + ⋯
 L(s)  = 1 − 2·9-s − 4/5·25-s − 2.85·49-s + 6.55·73-s + 3·81-s + 0.812·97-s + 1.81·121-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 + 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 + 0.0669·223-s + 8/5·225-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: $1$ Motivic weight: $$1$$ 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

 $$L(1)$$ $$\approx$$ $$1.03696$$ $$L(\frac12)$$ $$\approx$$ $$1.03696$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
good5$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
7$C_2$ $$( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2}$$
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$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
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$C_2^2$ $$( 1 - 94 T^{2} + p^{2} T^{4} )^{2}$$
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$C_2$ $$( 1 - 14 T + p T^{2} )^{4}$$
79$C_2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
83$C_2^2$ $$( 1 + 134 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 - p T^{2} )^{4}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$