# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s − 6·9-s − 40·13-s − 48·16-s + 340·25-s + 24·36-s − 520·37-s + 1.25e3·49-s + 160·52-s − 1.76e3·61-s + 448·64-s + 1.64e3·73-s − 693·81-s + 3.08e3·97-s − 1.36e3·100-s − 4.26e3·109-s + 240·117-s − 2.92e3·121-s + 127-s + 131-s + 137-s + 139-s + 288·144-s + 2.08e3·148-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 − 1/2·4-s − 2/9·9-s − 0.853·13-s − 3/4·16-s + 2.71·25-s + 1/9·36-s − 2.31·37-s + 3.65·49-s + 0.426·52-s − 3.71·61-s + 7/8·64-s + 2.62·73-s − 0.950·81-s + 3.22·97-s − 1.35·100-s − 3.74·109-s + 0.189·117-s − 2.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1/6·144-s + 1.15·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$20736$$    =    $$2^{8} \cdot 3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{12} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 20736,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$ $L(2)$ $\approx$ $0.650064$ $L(\frac12)$ $\approx$ $0.650064$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ 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$C_2^2$ $$1 + p^{2} T^{2} + p^{6} T^{4}$$
3$C_2^2$ $$1 + 2 p T^{2} + p^{6} T^{4}$$
good5$C_2^2$ $$( 1 - 34 p T^{2} + p^{6} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 626 T^{2} + p^{6} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 1462 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2$ $$( 1 + 10 T + p^{3} T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 8546 T^{2} + p^{6} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 8858 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 14926 T^{2} + p^{6} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 25658 T^{2} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 9122 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2$ $$( 1 + 130 T + p^{3} T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 122162 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 108554 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 170014 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 74 T^{2} + p^{6} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 380758 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2$ $$( 1 + 442 T + p^{3} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 60026 T^{2} + p^{6} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 364178 T^{2} + p^{6} T^{4} )^{2}$$
73$C_2$ $$( 1 - 410 T + p^{3} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 978818 T^{2} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 428954 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 703058 T^{2} + p^{6} T^{4} )^{2}$$
97$C_2$ $$( 1 - 770 T + p^{3} T^{2} )^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}