# Properties

 Degree 8 Conductor $2^{12} \cdot 3^{4}$ Sign $1$ Motivic weight 7 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 36·3-s − 160·4-s − 3.40e3·9-s + 5.76e3·12-s + 9.21e3·16-s + 4.62e4·19-s − 2.93e5·25-s + 2.12e5·27-s + 5.44e5·36-s + 1.98e6·43-s − 3.31e5·48-s − 1.50e6·49-s − 1.66e6·57-s + 1.14e6·64-s + 5.60e6·67-s − 8.89e6·73-s + 1.05e7·75-s − 7.40e6·76-s + 6.37e6·81-s + 2.74e7·97-s + 4.69e7·100-s − 3.40e7·108-s + 7.15e7·121-s + 127-s − 7.12e7·129-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 0.769·3-s − 5/4·4-s − 1.55·9-s + 0.962·12-s + 9/16·16-s + 1.54·19-s − 3.75·25-s + 2.08·27-s + 1.94·36-s + 3.79·43-s − 0.433·48-s − 1.82·49-s − 1.19·57-s + 0.546·64-s + 2.27·67-s − 2.67·73-s + 2.89·75-s − 1.93·76-s + 1.33·81-s + 3.05·97-s + 4.69·100-s − 2.60·108-s + 3.67·121-s − 2.92·129-s − 7/8·144-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$331776$$    =    $$2^{12} \cdot 3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : induced by $\chi_{24} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 331776,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)$ $L(4)$ $\approx$ $0.957440$ $L(\frac12)$ $\approx$ $0.957440$ $L(\frac{9}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ $$1 + 5 p^{5} T^{2} + p^{14} T^{4}$$
3$C_2$ $$( 1 + 2 p^{2} T + p^{7} T^{2} )^{2}$$
good5$C_2^2$ $$( 1 + 5866 p^{2} T^{2} + p^{14} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + 751570 T^{2} + p^{14} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 35789342 T^{2} + p^{14} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 1741490 p T^{2} + p^{14} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 61393730 T^{2} + p^{14} T^{4} )^{2}$$
19$C_2$ $$( 1 - 11570 T + p^{7} T^{2} )^{4}$$
23$C_2^2$ $$( 1 + 3725533390 T^{2} + p^{14} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 31499935018 T^{2} + p^{14} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 49885402622 T^{2} + p^{14} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 80259050 p^{2} T^{2} + p^{14} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 226584602162 T^{2} + p^{14} T^{4} )^{2}$$
43$C_2$ $$( 1 - 495062 T + p^{7} T^{2} )^{4}$$
47$C_2^2$ $$( 1 - 277104744290 T^{2} + p^{14} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 2021761791610 T^{2} + p^{14} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 2902252328702 T^{2} + p^{14} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 3094549289642 T^{2} + p^{14} T^{4} )^{2}$$
67$C_2$ $$( 1 - 1400126 T + p^{7} T^{2} )^{4}$$
71$C_2^2$ $$( 1 + 5449089705838 T^{2} + p^{14} T^{4} )^{2}$$
73$C_2$ $$( 1 + 2223598 T + p^{7} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 7153320805022 T^{2} + p^{14} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 45191963757710 T^{2} + p^{14} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 54294758858162 T^{2} + p^{14} T^{4} )^{2}$$
97$C_2$ $$( 1 - 6867926 T + p^{7} T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}