# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s − 16·7-s + 18·9-s + 68·13-s + 79·16-s + 96·21-s − 110·25-s + 198·27-s + 616·31-s − 1.15e3·37-s − 408·39-s + 548·43-s − 474·48-s + 128·49-s + 16·61-s − 288·63-s + 404·67-s − 2.51e3·73-s + 660·75-s − 1.27e3·81-s − 1.08e3·91-s − 3.69e3·93-s + 1.90e3·97-s − 6.83e3·103-s + 6.93e3·111-s − 1.26e3·112-s + 1.22e3·117-s + ⋯
 L(s)  = 1 − 1.15·3-s − 0.863·7-s + 2/3·9-s + 1.45·13-s + 1.23·16-s + 0.997·21-s − 0.879·25-s + 1.41·27-s + 3.56·31-s − 5.13·37-s − 1.67·39-s + 1.94·43-s − 1.42·48-s + 0.373·49-s + 0.0335·61-s − 0.575·63-s + 0.736·67-s − 4.02·73-s + 1.01·75-s − 1.75·81-s − 1.25·91-s − 4.12·93-s + 1.99·97-s − 6.53·103-s + 5.93·111-s − 1.06·112-s + 0.967·117-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$3^{8} \cdot 5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{15} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 3^{8} \cdot 5^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$ $L(2)$ $\approx$ $0.602084$ $L(\frac12)$ $\approx$ $0.602084$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $$F_p(T)$$ is a polynomial of degree 16. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 $$1 + 2 p T + 2 p^{2} T^{2} - 22 p^{2} T^{3} - 158 p^{2} T^{4} - 22 p^{5} T^{5} + 2 p^{8} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8}$$
5 $$1 + 22 p T^{2} + 42 p^{3} T^{4} + 22 p^{7} T^{6} + p^{12} T^{8}$$
good2 $$1 - 79 T^{4} + 39 p^{6} T^{8} - 79 p^{12} T^{12} + p^{24} T^{16}$$
7 $$( 1 + 8 T + 32 T^{2} + 744 T^{3} - 45202 T^{4} + 744 p^{3} T^{5} + 32 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
11 $$( 1 - 3334 T^{2} + 6292986 T^{4} - 3334 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
13 $$( 1 - 34 T + 578 T^{2} - 77418 T^{3} + 10363058 T^{4} - 77418 p^{3} T^{5} + 578 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
17 $$1 - 10588864 T^{4} - 447530075229954 T^{8} - 10588864 p^{12} T^{12} + p^{24} T^{16}$$
19 $$( 1 - 24664 T^{2} + 245125086 T^{4} - 24664 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
23 $$1 + 232150736 T^{4} + 54876630002867166 T^{8} + 232150736 p^{12} T^{12} + p^{24} T^{16}$$
29 $$( 1 + 79166 T^{2} + 2712313506 T^{4} + 79166 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
31 $$( 1 - 154 T + 36486 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
37 $$( 1 + 578 T + 167042 T^{2} + 53079474 T^{3} + 15170807378 T^{4} + 53079474 p^{3} T^{5} + 167042 p^{6} T^{6} + 578 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
41 $$( 1 - 115444 T^{2} + 7614039366 T^{4} - 115444 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
43 $$( 1 - 274 T + 37538 T^{2} - 17976318 T^{3} + 8415346898 T^{4} - 17976318 p^{3} T^{5} + 37538 p^{6} T^{6} - 274 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
47 $$1 + 29984206736 T^{4} +$$$$42\!\cdots\!06$$$$T^{8} + 29984206736 p^{12} T^{12} + p^{24} T^{16}$$
53 $$1 - 24371904064 T^{4} +$$$$31\!\cdots\!06$$$$T^{8} - 24371904064 p^{12} T^{12} + p^{24} T^{16}$$
59 $$( 1 + 112106 T^{2} + 56049165066 T^{4} + 112106 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
61 $$( 1 - 2 T + p^{3} T^{2} )^{8}$$
67 $$( 1 - 202 T + 20402 T^{2} - 3297246 T^{3} - 80373232942 T^{4} - 3297246 p^{3} T^{5} + 20402 p^{6} T^{6} - 202 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
71 $$( 1 - 863584 T^{2} + 377860054206 T^{4} - 863584 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
73 $$( 1 + 1256 T + 788768 T^{2} + 733362072 T^{3} + 643873740638 T^{4} + 733362072 p^{3} T^{5} + 788768 p^{6} T^{6} + 1256 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
79 $$( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 1624084 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
83 $$1 + 1005705244496 T^{4} +$$$$46\!\cdots\!26$$$$T^{8} + 1005705244496 p^{12} T^{12} + p^{24} T^{16}$$
89 $$( 1 - 806584 T^{2} + 1118488022286 T^{4} - 806584 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
97 $$( 1 - 952 T + 453152 T^{2} - 121725576 T^{3} - 583228842562 T^{4} - 121725576 p^{3} T^{5} + 453152 p^{6} T^{6} - 952 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}