Properties

 Label 8-320e4-1.1-c1e4-0-3 Degree $8$ Conductor $10485760000$ Sign $1$ Analytic cond. $42.6293$ Root an. cond. $1.59850$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 4·5-s − 4·13-s + 4·17-s + 2·25-s − 20·37-s + 8·41-s + 28·53-s − 24·61-s − 16·65-s − 28·73-s + 18·81-s + 16·85-s − 28·97-s + 40·101-s + 36·113-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + ⋯
 L(s)  = 1 + 1.78·5-s − 1.10·13-s + 0.970·17-s + 2/5·25-s − 3.28·37-s + 1.24·41-s + 3.84·53-s − 3.07·61-s − 1.98·65-s − 3.27·73-s + 2·81-s + 1.73·85-s − 2.84·97-s + 3.98·101-s + 3.38·113-s + 1.81·121-s − 2.50·125-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 + 8/13·169-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$42.6293$$ Root analytic conductor: $$1.59850$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{320} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$2.460927437$$ $$L(\frac12)$$ $$\approx$$ $$2.460927437$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
good3$C_2$$\times$$C_2$ $$( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2}$$
7$C_2^3$ $$1 - 34 T^{4} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}$$
17$C_2^2$ $$( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 + 542 T^{4} + p^{4} T^{8}$$
29$C_2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
31$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
43$C_2^3$ $$1 + 2702 T^{4} + p^{4} T^{8}$$
47$C_2^3$ $$1 + 3326 T^{4} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 70 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
67$C_2^3$ $$1 - 2578 T^{4} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - 34 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2^3$ $$1 + 2606 T^{4} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 114 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$