# Properties

 Degree 8 Conductor $2^{12}$ Sign $1$ Motivic weight 6 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 48·3-s − 20·4-s − 96·6-s + 40·8-s − 636·9-s + 976·11-s + 960·12-s − 3.28e3·16-s + 4.16e3·17-s − 1.27e3·18-s − 1.45e3·19-s + 1.95e3·22-s − 1.92e3·24-s + 1.93e4·25-s + 7.99e4·27-s − 1.82e4·32-s − 4.68e4·33-s + 8.33e3·34-s + 1.27e4·36-s − 2.91e3·38-s − 1.17e5·41-s + 1.97e5·43-s − 1.95e4·44-s + 1.57e5·48-s + 2.36e5·49-s + 3.86e4·50-s + ⋯
 L(s)  = 1 + 1/4·2-s − 1.77·3-s − 0.312·4-s − 4/9·6-s + 5/64·8-s − 0.872·9-s + 0.733·11-s + 5/9·12-s − 0.800·16-s + 0.848·17-s − 0.218·18-s − 0.212·19-s + 0.183·22-s − 0.138·24-s + 1.23·25-s + 4.06·27-s − 0.557·32-s − 1.30·33-s + 0.212·34-s + 0.272·36-s − 0.0530·38-s − 1.71·41-s + 2.48·43-s − 0.229·44-s + 1.42·48-s + 2.00·49-s + 0.308·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$4096$$    =    $$2^{12}$$ $$\varepsilon$$ = $1$ motivic weight = $$6$$ character : induced by $\chi_{8} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 4096,\ (\ :3, 3, 3, 3),\ 1)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$0.589912$$ $$L(\frac12)$$ $$\approx$$ $$0.589912$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 8. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$D_{4}$ $$1 - p T + 3 p^{3} T^{2} - p^{7} T^{3} + p^{12} T^{4}$$
good3$D_{4}$ $$( 1 + 8 p T + 394 p T^{2} + 8 p^{7} T^{3} + p^{12} T^{4} )^{2}$$
5$C_2^2 \wr C_2$ $$1 - 772 p^{2} T^{2} + 2049246 p^{3} T^{4} - 772 p^{14} T^{6} + p^{24} T^{8}$$
7$C_2^2 \wr C_2$ $$1 - 236356 T^{2} + 28021959366 T^{4} - 236356 p^{12} T^{6} + p^{24} T^{8}$$
11$D_{4}$ $$( 1 - 488 T + 2238078 T^{2} - 488 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
13$C_2^2 \wr C_2$ $$1 - 4994596 T^{2} + 30260873415846 T^{4} - 4994596 p^{12} T^{6} + p^{24} T^{8}$$
17$D_{4}$ $$( 1 - 2084 T + 32560902 T^{2} - 2084 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
19$D_{4}$ $$( 1 + 728 T + 92449758 T^{2} + 728 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 - 212117956 T^{2} + 1001159632705962 p T^{4} - 212117956 p^{12} T^{6} + p^{24} T^{8}$$
29$C_2^2 \wr C_2$ $$1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - 1409719204 p^{12} T^{6} + p^{24} T^{8}$$
31$C_2^2 \wr C_2$ $$1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - 2758360324 p^{12} T^{6} + p^{24} T^{8}$$
37$C_2^2 \wr C_2$ $$1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - 8121202276 p^{12} T^{6} + p^{24} T^{8}$$
41$D_{4}$ $$( 1 + 58972 T + 240432438 p T^{2} + 58972 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 2296 p T + 13190101374 T^{2} - 2296 p^{7} T^{3} + p^{12} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 - 13578767236 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} - 13578767236 p^{12} T^{6} + p^{24} T^{8}$$
53$C_2^2 \wr C_2$ $$1 - 42194545636 T^{2} +$$$$11\!\cdots\!86$$$$T^{4} - 42194545636 p^{12} T^{6} + p^{24} T^{8}$$
59$D_{4}$ $$( 1 - 271016 T + 102678575166 T^{2} - 271016 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
61$C_2^2 \wr C_2$ $$1 - 2637195124 p T^{2} +$$$$11\!\cdots\!46$$$$T^{4} - 2637195124 p^{13} T^{6} + p^{24} T^{8}$$
67$D_{4}$ $$( 1 + 395096 T + 204987839262 T^{2} + 395096 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 - 164364621124 T^{2} +$$$$21\!\cdots\!06$$$$T^{4} - 164364621124 p^{12} T^{6} + p^{24} T^{8}$$
73$D_{4}$ $$( 1 - 221956 T + 284381984742 T^{2} - 221956 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
79$C_2^2 \wr C_2$ $$1 - 828257339524 T^{2} +$$$$28\!\cdots\!06$$$$T^{4} - 828257339524 p^{12} T^{6} + p^{24} T^{8}$$
83$D_{4}$ $$( 1 + 1732504 T + 1400265667422 T^{2} + 1732504 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
89$D_{4}$ $$( 1 - 380612 T + 970422538278 T^{2} - 380612 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 463388 T + 953987784774 T^{2} + 463388 p^{6} T^{3} + p^{12} T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}