# 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 − 8·2-s + 36·4-s − 120·8-s − 8·13-s + 330·16-s + 8·23-s + 64·26-s − 792·32-s + 8·41-s − 64·46-s − 2·49-s − 288·52-s + 8·53-s + 1.71e3·64-s + 48·73-s − 8·79-s − 64·82-s − 8·89-s + 288·92-s − 24·97-s + 16·98-s − 8·101-s − 56·103-s + 960·104-s − 64·106-s − 40·107-s + 32·109-s + ⋯
 L(s)  = 1 − 5.65·2-s + 18·4-s − 42.4·8-s − 2.21·13-s + 82.5·16-s + 1.66·23-s + 12.5·26-s − 140.·32-s + 1.24·41-s − 9.43·46-s − 2/7·49-s − 39.9·52-s + 1.09·53-s + 214.5·64-s + 5.61·73-s − 0.900·79-s − 7.06·82-s − 0.847·89-s + 30.0·92-s − 2.43·97-s + 1.61·98-s − 0.796·101-s − 5.51·103-s + 94.1·104-s − 6.21·106-s − 3.86·107-s + 3.06·109-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 = $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $0.05166075196$ $L(\frac12)$ $\approx$ $0.05166075196$ $L(\frac{3}{2})$ 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 )^{8}$$
3 $$1$$
5 $$1$$
7 $$1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8}$$
good11 $$1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 11916 p^{2} T^{10} + 776 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16}$$
13 $$( 1 + 4 T + 30 T^{2} + 132 T^{3} + 514 T^{4} + 132 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
17 $$1 - 36 T^{2} + 248 T^{4} + 5364 T^{6} - 133842 T^{8} + 5364 p^{2} T^{10} + 248 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16}$$
19 $$1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 31140 p^{2} T^{10} + 1256 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16}$$
23 $$( 1 - 4 T + 48 T^{2} - 180 T^{3} + 1438 T^{4} - 180 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
29 $$1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 399744 p^{2} T^{10} + 8732 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16}$$
31 $$1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21984 p^{2} T^{10} + 572 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16}$$
37 $$1 - 152 T^{2} + 10004 T^{4} - 394632 T^{6} + 13477382 T^{8} - 394632 p^{2} T^{10} + 10004 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16}$$
41 $$( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 468 p T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
43 $$1 - 228 T^{2} + 23256 T^{4} - 1476524 T^{6} + 70398222 T^{8} - 1476524 p^{2} T^{10} + 23256 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 + 68 T^{2} + 10040 T^{4} + 9492 p T^{6} + 34404910 T^{8} + 9492 p^{3} T^{10} + 10040 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16}$$
53 $$( 1 - 4 T + 160 T^{2} - 12 p T^{3} + 11630 T^{4} - 12 p^{2} T^{5} + 160 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
59 $$( 1 + 42 T^{2} + 312 T^{3} + 3106 T^{4} + 312 p T^{5} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 2804772 p^{2} T^{10} + 31208 p^{4} T^{12} - 236 p^{6} T^{14} + p^{8} T^{16}$$
67 $$1 - 180 T^{2} + 22904 T^{4} - 1953756 T^{6} + 148114446 T^{8} - 1953756 p^{2} T^{10} + 22904 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16}$$
71 $$1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 12828132 p^{2} T^{10} + 98600 p^{4} T^{12} - 460 p^{6} T^{14} + p^{8} T^{16}$$
73 $$( 1 - 24 T + 404 T^{2} - 4680 T^{3} + 45878 T^{4} - 4680 p T^{5} + 404 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
79 $$( 1 + 4 T + 48 T^{2} + 532 T^{3} + 9470 T^{4} + 532 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$1 - 424 T^{2} + 86492 T^{4} - 11335896 T^{6} + 1081357798 T^{8} - 11335896 p^{2} T^{10} + 86492 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16}$$
89 $$( 1 + 4 T + 118 T^{2} - 1308 T^{3} - 670 T^{4} - 1308 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$( 1 + 12 T + 176 T^{2} + 2148 T^{3} + 29438 T^{4} + 2148 p T^{5} + 176 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}