# Properties

 Degree 8 Conductor $2^{12} \cdot 3^{4}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 2·4-s − 8·8-s + 6·9-s − 32·11-s − 4·16-s − 8·17-s + 12·18-s + 32·19-s − 64·22-s + 28·25-s − 8·32-s − 16·34-s − 12·36-s + 64·38-s + 40·41-s + 32·43-s + 64·44-s + 76·49-s + 56·50-s − 128·59-s − 24·64-s − 256·67-s + 16·68-s − 48·72-s + 200·73-s − 64·76-s + ⋯
 L(s)  = 1 + 2-s − 1/2·4-s − 8-s + 2/3·9-s − 2.90·11-s − 1/4·16-s − 0.470·17-s + 2/3·18-s + 1.68·19-s − 2.90·22-s + 1.11·25-s − 1/4·32-s − 0.470·34-s − 1/3·36-s + 1.68·38-s + 0.975·41-s + 0.744·43-s + 1.45·44-s + 1.55·49-s + 1.11·50-s − 2.16·59-s − 3/8·64-s − 3.82·67-s + 4/17·68-s − 2/3·72-s + 2.73·73-s − 0.842·76-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$331776$$    =    $$2^{12} \cdot 3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{24} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 331776,\ (\ :1, 1, 1, 1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.883563$ $L(\frac12)$ $\approx$ $0.883563$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$D_{4}$ $$1 - p T + 3 p T^{2} - p^{3} T^{3} + p^{4} T^{4}$$
3$C_2$ $$( 1 - p T^{2} )^{2}$$
good5$C_2^2 \wr C_2$ $$1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8}$$
7$C_2^2 \wr C_2$ $$1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8}$$
11$C_2$ $$( 1 + 8 T + p^{2} T^{2} )^{4}$$
13$C_2^2 \wr C_2$ $$1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8}$$
17$D_{4}$ $$( 1 + 4 T + 390 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$D_{4}$ $$( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8}$$
29$C_2^2 \wr C_2$ $$1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8}$$
31$C_2^2 \wr C_2$ $$1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8}$$
37$C_2^2 \wr C_2$ $$1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8}$$
41$D_{4}$ $$( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8}$$
53$C_2^2 \wr C_2$ $$1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8}$$
59$D_{4}$ $$( 1 + 64 T + 7554 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
61$C_2^2 \wr C_2$ $$1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8}$$
67$D_{4}$ $$( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$C_2^2 \wr C_2$ $$1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8}$$
83$D_{4}$ $$( 1 - 80 T + 14610 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
89$D_{4}$ $$( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$D_{4}$ $$( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}