Properties

 Degree $8$ Conductor $8.362\times 10^{13}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 24·19-s − 4·25-s + 12·31-s + 16·37-s + 14·49-s − 60·103-s + 20·109-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 + ⋯
 L(s)  = 1 + 5.50·19-s − 4/5·25-s + 2.15·31-s + 2.63·37-s + 2·49-s − 5.91·103-s + 1.91·109-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 + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \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^{16} \cdot 3^{12} \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: $$2^{16} \cdot 3^{12} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{3024} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

Particular Values

 $$L(1)$$ $$\approx$$ $$8.881245432$$ $$L(\frac12)$$ $$\approx$$ $$8.881245432$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
good5$C_2^3$ $$1 + 4 T^{2} - 9 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^3$ $$1 - 20 T^{2} + 279 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2$ $$( 1 - p T^{2} )^{4}$$
17$C_2^2$ $$( 1 + 20 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 + 4 T^{2} - 513 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^3$ $$1 - 68 T^{2} + 2943 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2$ $$( 1 - p T^{2} )^{4}$$
47$C_2$ $$( 1 + p T^{2} )^{4}$$
53$C_2$ $$( 1 + p T^{2} )^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{4}$$
61$C_2$ $$( 1 - p T^{2} )^{4}$$
67$C_2$ $$( 1 - p T^{2} )^{4}$$
71$C_2^3$ $$1 + 100 T^{2} + 4959 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2$ $$( 1 - p T^{2} )^{4}$$
79$C_2$ $$( 1 - p T^{2} )^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2^3$ $$1 + 172 T^{2} + 21663 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2$ $$( 1 - p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$