Properties

 Degree $8$ Conductor $1.216\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·4-s + 4·7-s + 3·16-s − 8·28-s + 8·37-s − 16·43-s − 2·49-s − 4·64-s − 32·67-s − 40·79-s − 9·81-s + 40·109-s + 12·112-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 32·172-s + 173-s + ⋯
 L(s)  = 1 − 4-s + 1.51·7-s + 3/4·16-s − 1.51·28-s + 1.31·37-s − 2.43·43-s − 2/7·49-s − 1/2·64-s − 3.90·67-s − 4.50·79-s − 81-s + 3.83·109-s + 1.13·112-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 2.43·172-s + 0.0760·173-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1050} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.204383746$$ $$L(\frac12)$$ $$\approx$$ $$1.204383746$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_2^2$ $$1 + p^{2} T^{4}$$
5 $$1$$
7$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
good11$C_2$ $$( 1 - p T^{2} )^{4}$$
13$C_2^2$ $$( 1 - 20 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 10 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 32 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - p T^{2} )^{4}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
41$C_2^2$ $$( 1 + 58 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
47$C_2^2$ $$( 1 + 70 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 32 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 28 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
71$C_2$ $$( 1 - p T^{2} )^{4}$$
73$C_2$ $$( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 + 160 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 + p T^{2} )^{4}$$
97$C_2^2$ $$( 1 - 170 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$