Properties

 Degree 16 Conductor $2^{24} \cdot 3^{8} \cdot 7^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·4-s + 8·7-s + 8·9-s − 5·16-s + 32·25-s − 16·28-s − 16·36-s + 12·49-s + 64·63-s + 20·64-s − 80·79-s + 30·81-s − 64·100-s − 40·112-s − 40·121-s + 127-s + 131-s + 137-s + 139-s − 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + ⋯
 L(s)  = 1 − 4-s + 3.02·7-s + 8/3·9-s − 5/4·16-s + 32/5·25-s − 3.02·28-s − 8/3·36-s + 12/7·49-s + 8.06·63-s + 5/2·64-s − 9.00·79-s + 10/3·81-s − 6.39·100-s − 3.77·112-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.33·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-s + 0.0760·173-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{24} \cdot 3^{8} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{168} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(16,\ 2^{24} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$2.89364$$ $$L(\frac12)$$ $$\approx$$ $$2.89364$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$( 1 + T^{2} + p^{2} T^{4} )^{2}$$
3 $$( 1 - 4 T^{2} + p^{2} T^{4} )^{2}$$
7 $$( 1 - 2 T + p T^{2} )^{4}$$
good5 $$( 1 - 8 T^{2} + p^{2} T^{4} )^{4}$$
11 $$( 1 + 10 T^{2} + p^{2} T^{4} )^{4}$$
13 $$( 1 + 16 T^{2} + p^{2} T^{4} )^{4}$$
17 $$( 1 + p T^{2} )^{8}$$
19 $$( 1 + 28 T^{2} + p^{2} T^{4} )^{4}$$
23 $$( 1 - 26 T^{2} + p^{2} T^{4} )^{4}$$
29 $$( 1 + 10 T^{2} + p^{2} T^{4} )^{4}$$
31 $$( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4}$$
37 $$( 1 - p T^{2} )^{8}$$
41 $$( 1 - 38 T^{2} + p^{2} T^{4} )^{4}$$
43 $$( 1 - 26 T^{2} + p^{2} T^{4} )^{4}$$
47 $$( 1 - 26 T^{2} + p^{2} T^{4} )^{4}$$
53 $$( 1 + p T^{2} )^{8}$$
59 $$( 1 - 20 T^{2} + p^{2} T^{4} )^{4}$$
61 $$( 1 + 112 T^{2} + p^{2} T^{4} )^{4}$$
67 $$( 1 - 74 T^{2} + p^{2} T^{4} )^{4}$$
71 $$( 1 - 62 T^{2} + p^{2} T^{4} )^{4}$$
73 $$( 1 + 70 T^{2} + p^{2} T^{4} )^{4}$$
79 $$( 1 + 10 T + p T^{2} )^{8}$$
83 $$( 1 - 116 T^{2} + p^{2} T^{4} )^{4}$$
89 $$( 1 + p T^{2} )^{8}$$
97 $$( 1 - 170 T^{2} + p^{2} T^{4} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}