# Properties

 Degree 8 Conductor $2^{8} \cdot 5^{4}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 16·9-s − 16·13-s − 24·17-s − 32·18-s + 10·25-s + 32·26-s − 8·29-s + 32·32-s + 48·34-s + 16·37-s − 112·41-s + 96·49-s − 20·50-s − 176·53-s + 16·58-s + 128·61-s − 64·64-s + 264·73-s − 32·74-s + 50·81-s + 224·82-s − 88·89-s − 264·97-s − 192·98-s + 328·101-s + 352·106-s + ⋯
 L(s)  = 1 − 2-s + 16/9·9-s − 1.23·13-s − 1.41·17-s − 1.77·18-s + 2/5·25-s + 1.23·26-s − 0.275·29-s + 32-s + 1.41·34-s + 0.432·37-s − 2.73·41-s + 1.95·49-s − 2/5·50-s − 3.32·53-s + 8/29·58-s + 2.09·61-s − 64-s + 3.61·73-s − 0.432·74-s + 0.617·81-s + 2.73·82-s − 0.988·89-s − 2.72·97-s − 1.95·98-s + 3.24·101-s + 3.32·106-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$160000$$    =    $$2^{8} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{20} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 160000,\ (\ :1, 1, 1, 1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.342682$ $L(\frac12)$ $\approx$ $0.342682$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_4$ $$1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4}$$
5$C_2$ $$( 1 - p T^{2} )^{2}$$
good3$C_2^2:C_4$ $$1 - 16 T^{2} + 206 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8}$$
7$C_2^2:C_4$ $$1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8}$$
11$C_2^2:C_4$ $$1 - 84 T^{2} - 7674 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8}$$
13$D_{4}$ $$( 1 + 8 T + 334 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
17$D_{4}$ $$( 1 + 12 T + 294 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$C_2^2:C_4$ $$1 - 1124 T^{2} + 571366 T^{4} - 1124 p^{4} T^{6} + p^{8} T^{8}$$
23$C_2^2:C_4$ $$1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8}$$
29$D_{4}$ $$( 1 + 4 T + 1606 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
31$C_2^2:C_4$ $$1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 - 8 T + 2254 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_{4}$ $$( 1 + 56 T + 3966 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$C_2^2:C_4$ $$1 - 6896 T^{2} + 18665806 T^{4} - 6896 p^{4} T^{6} + p^{8} T^{8}$$
47$C_2^2:C_4$ $$1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 + 88 T + 7054 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$C_2^2:C_4$ $$1 - 8164 T^{2} + 34261926 T^{4} - 8164 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 - 64 T + 5086 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$C_2^2:C_4$ $$1 - 7536 T^{2} + 47276046 T^{4} - 7536 p^{4} T^{6} + p^{8} T^{8}$$
71$C_2^2:C_4$ $$1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 - 132 T + 9894 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$C_2^2:C_4$ $$1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8}$$
83$C_2^2:C_4$ $$1 - 21296 T^{2} + 201120526 T^{4} - 21296 p^{4} T^{6} + p^{8} T^{8}$$
89$D_{4}$ $$( 1 + 44 T + 8326 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}