# 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 − 4·2-s + 6·4-s − 15·16-s + 24·17-s + 24·19-s − 12·31-s + 24·32-s − 96·34-s − 96·38-s − 24·47-s − 10·49-s + 12·53-s + 24·61-s + 48·62-s − 6·64-s + 144·68-s + 144·76-s − 28·79-s + 96·94-s + 40·98-s − 48·106-s + 12·107-s + 40·109-s − 26·121-s − 96·122-s − 72·124-s + 127-s + ⋯
 L(s)  = 1 − 2.82·2-s + 3·4-s − 3.75·16-s + 5.82·17-s + 5.50·19-s − 2.15·31-s + 4.24·32-s − 16.4·34-s − 15.5·38-s − 3.50·47-s − 1.42·49-s + 1.64·53-s + 3.07·61-s + 6.09·62-s − 3/4·64-s + 17.4·68-s + 16.5·76-s − 3.15·79-s + 9.90·94-s + 4.04·98-s − 4.66·106-s + 1.16·107-s + 3.83·109-s − 2.36·121-s − 8.69·122-s − 6.46·124-s + 0.0887·127-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 = $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $3.165757436$ $L(\frac12)$ $\approx$ $3.165757436$ $L(\frac{3}{2})$ 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 + T^{2} )^{4}$$
3 $$1$$
5 $$1$$
7 $$1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}$$
good11 $$( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
13 $$( 1 + 20 T^{2} + p^{2} T^{4} )^{4}$$
17 $$( 1 - 12 T + 88 T^{2} - 480 T^{3} + 2127 T^{4} - 480 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
19 $$( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2487 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
23 $$( 1 - 28 T^{2} + 255 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
29 $$( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 66 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
37 $$1 + 80 T^{2} + 3214 T^{4} + 35840 T^{6} - 485165 T^{8} + 35840 p^{2} T^{10} + 3214 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16}$$
41 $$( 1 + 20 T^{2} - 1146 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
43 $$( 1 - 104 T^{2} + 5250 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
47 $$( 1 + 12 T + 148 T^{2} + 1200 T^{3} + 10047 T^{4} + 1200 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
53 $$( 1 - 6 T - 61 T^{2} + 54 T^{3} + 4692 T^{4} + 54 p T^{5} - 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
59 $$1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 30058 p^{2} T^{10} - 4727 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16}$$
61 $$( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 1536 p T^{5} + 176 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
71 $$( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$1 - 220 T^{2} + 26794 T^{4} - 2408560 T^{6} + 180497395 T^{8} - 2408560 p^{2} T^{10} + 26794 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 434 p T^{5} + 7 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$( 1 - 278 T^{2} + 32811 T^{4} - 278 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 + 190 T^{2} + 20643 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}