# Properties

 Degree 16 Conductor $2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 10·16-s + 8·49-s − 20·64-s + 32·79-s + 80·109-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 32·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 − 2·4-s + 5/2·16-s + 8/7·49-s − 5/2·64-s + 3.60·79-s + 7.66·109-s + 7.27·121-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 + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2.28·196-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^{8} \cdot 3^{16} \cdot 5^{16} \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^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{3150} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $8.726549331$ $L(\frac12)$ $\approx$ $8.726549331$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$( 1 + T^{2} )^{4}$$
3 $$1$$
5 $$1$$
7 $$( 1 - 4 T^{2} + p^{2} T^{4} )^{2}$$
good11 $$( 1 - 20 T^{2} + p^{2} T^{4} )^{4}$$
13 $$( 1 - 16 T^{2} + p^{2} T^{4} )^{4}$$
17 $$( 1 + 14 T^{2} + p^{2} T^{4} )^{4}$$
19 $$( 1 - p T^{2} )^{8}$$
23 $$( 1 - 10 T^{2} + p^{2} T^{4} )^{4}$$
29 $$( 1 - 50 T^{2} + p^{2} T^{4} )^{4}$$
31 $$( 1 - p T^{2} )^{8}$$
37 $$( 1 + 56 T^{2} + p^{2} T^{4} )^{4}$$
41 $$( 1 - 8 T^{2} + p^{2} T^{4} )^{4}$$
43 $$( 1 + 14 T^{2} + p^{2} T^{4} )^{4}$$
47 $$( 1 + 74 T^{2} + p^{2} T^{4} )^{4}$$
53 $$( 1 - 70 T^{2} + p^{2} T^{4} )^{4}$$
59 $$( 1 + 28 T^{2} + p^{2} T^{4} )^{4}$$
61 $$( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4}$$
67 $$( 1 + p T^{2} )^{8}$$
71 $$( 1 - 110 T^{2} + p^{2} T^{4} )^{4}$$
73 $$( 1 - 106 T^{2} + p^{2} T^{4} )^{4}$$
79 $$( 1 - 4 T + p T^{2} )^{8}$$
83 $$( 1 + 86 T^{2} + p^{2} T^{4} )^{4}$$
89 $$( 1 + 88 T^{2} + p^{2} T^{4} )^{4}$$
97 $$( 1 - 34 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}