Properties

 Label 8-1200e4-1.1-c1e4-0-7 Degree $8$ Conductor $2.074\times 10^{12}$ Sign $1$ Analytic cond. $8430.11$ Root an. cond. $3.09548$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 4·3-s + 6·9-s − 24·23-s − 4·27-s + 24·47-s + 28·49-s − 8·61-s − 96·69-s − 37·81-s − 24·83-s − 72·107-s + 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 2.30·3-s + 2·9-s − 5.00·23-s − 0.769·27-s + 3.50·47-s + 4·49-s − 1.02·61-s − 11.5·69-s − 4.11·81-s − 2.63·83-s − 6.96·107-s + 0.766·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \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^{16} \cdot 3^{4} \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: $$2^{16} \cdot 3^{4} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$8430.11$$ Root analytic conductor: $$3.09548$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.641535451$$ $$L(\frac12)$$ $$\approx$$ $$1.641535451$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
5 $$1$$
good7$C_2$ $$( 1 - p T^{2} )^{4}$$
11$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
29$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
53$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 94 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
67$C_2$ $$( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}$$
71$C_2^2$ $$( 1 + 46 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
89$C_2$ $$( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$