Properties

 Label 8-63e8-1.1-c1e4-0-6 Degree $8$ Conductor $2.482\times 10^{14}$ Sign $1$ Analytic cond. $1.00886\times 10^{6}$ Root an. cond. $5.62962$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

Origins of factors

Dirichlet series

 L(s)  = 1 − 3·4-s + 4·16-s − 20·25-s − 12·37-s − 24·43-s − 9·64-s − 8·67-s − 16·79-s + 60·100-s − 72·109-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 72·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 3/2·4-s + 16-s − 4·25-s − 1.97·37-s − 3.65·43-s − 9/8·64-s − 0.977·67-s − 1.80·79-s + 6·100-s − 6.89·109-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 5.48·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$3^{16} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$1.00886\times 10^{6}$$ Root analytic conductor: $$5.62962$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3969} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

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)$
bad3 $$1$$
7 $$1$$
good2$C_2^3$ $$1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8}$$
5$C_2$ $$( 1 + p T^{2} )^{4}$$
11$C_2^3$ $$1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8}$$
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$C_2^2$ $$( 1 + 18 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 54 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2$ $$( 1 + p T^{2} )^{4}$$
53$C_2^3$ $$1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2$ $$( 1 + p T^{2} )^{4}$$
61$C_2$ $$( 1 + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^3$ $$1 - 114 T^{2} + 7955 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2$ $$( 1 + p T^{2} )^{4}$$
79$C_2^2$ $$( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{4}$$
97$C_2$ $$( 1 + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$