# 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 + 4·7-s + 10·16-s − 16·28-s + 8·37-s − 8·41-s + 16·43-s + 40·47-s + 10·49-s − 20·64-s − 32·67-s + 8·79-s − 16·83-s − 8·89-s + 8·101-s − 32·109-s + 40·112-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + ⋯
 L(s)  = 1 − 2·4-s + 1.51·7-s + 5/2·16-s − 3.02·28-s + 1.31·37-s − 1.24·41-s + 2.43·43-s + 5.83·47-s + 10/7·49-s − 5/2·64-s − 3.90·67-s + 0.900·79-s − 1.75·83-s − 0.847·89-s + 0.796·101-s − 3.06·109-s + 3.77·112-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-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 = $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $0.8265720314$ $L(\frac12)$ $\approx$ $0.8265720314$ $L(\frac{3}{2})$ 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 + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + 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 - 44 T^{2} + 872 T^{4} - 756 p T^{6} + 102190 T^{8} - 756 p^{3} T^{10} + 872 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16}$$
17 $$( 1 + 18 T^{2} - 120 T^{3} - 38 T^{4} - 120 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
19 $$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 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 123312 p^{2} T^{10} + 3740 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16}$$
29 $$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 - 4 T + 84 T^{2} - 532 T^{3} + 3602 T^{4} - 532 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
41 $$( 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 - 8 T + 146 T^{2} - 984 T^{3} + 8842 T^{4} - 984 p T^{5} + 146 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
47 $$( 1 - 20 T + 166 T^{2} - 396 T^{3} - 838 T^{4} - 396 p T^{5} + 166 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
53 $$1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 3946320 p^{2} T^{10} + 43772 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 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 + 16 T + 218 T^{2} + 1728 T^{3} + 15338 T^{4} + 1728 p T^{5} + 218 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
71 $$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 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 3074136 p^{2} T^{10} + 30332 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 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 + 8 T + 244 T^{2} + 1800 T^{3} + 27878 T^{4} + 1800 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
89 $$( 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 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 4059696 p^{2} T^{10} + 38300 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}