# Properties

 Degree $4$ Conductor $160000$ Sign $1$ Motivic weight $5$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 20·3-s + 200·7-s + 55·9-s + 196·11-s + 360·13-s − 1.49e3·17-s + 3.18e3·19-s + 4.00e3·21-s + 1.56e3·23-s − 940·27-s − 3.92e3·29-s + 1.09e3·31-s + 3.92e3·33-s − 2.02e3·37-s + 7.20e3·39-s + 2.77e4·41-s − 3.00e3·43-s + 2.57e4·47-s − 2.65e3·49-s − 2.98e4·51-s + 2.69e4·53-s + 6.36e4·57-s − 1.19e4·59-s − 2.43e4·61-s + 1.10e4·63-s − 4.00e4·67-s + 3.12e4·69-s + ⋯
 L(s)  = 1 + 1.28·3-s + 1.54·7-s + 0.226·9-s + 0.488·11-s + 0.590·13-s − 1.25·17-s + 2.02·19-s + 1.97·21-s + 0.614·23-s − 0.248·27-s − 0.865·29-s + 0.204·31-s + 0.626·33-s − 0.242·37-s + 0.758·39-s + 2.57·41-s − 0.247·43-s + 1.70·47-s − 0.157·49-s − 1.60·51-s + 1.31·53-s + 2.59·57-s − 0.447·59-s − 0.839·61-s + 0.349·63-s − 1.09·67-s + 0.788·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$160000$$    =    $$2^{8} \cdot 5^{4}$$ Sign: $1$ Motivic weight: $$5$$ Character: induced by $\chi_{400} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 160000,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$8.728292643$$ $$L(\frac12)$$ $$\approx$$ $$8.728292643$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3$D_{4}$ $$1 - 20 T + 115 p T^{2} - 20 p^{5} T^{3} + p^{10} T^{4}$$
7$D_{4}$ $$1 - 200 T + 42650 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 196 T + 181081 T^{2} - 196 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 360 T + 713290 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 1490 T + 2280355 T^{2} + 1490 p^{5} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 - 1560 T + 13472410 T^{2} - 1560 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 3920 T + 44478298 T^{2} + 3920 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 1096 T - 15343894 T^{2} - 1096 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 + 2020 T + 130823790 T^{2} + 2020 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 3000 T + 23431750 T^{2} + 3000 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 25760 T + 480952270 T^{2} - 25760 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 26980 T + 984299470 T^{2} - 26980 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 11960 T + 1234152598 T^{2} + 11960 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 + 40060 T + 2949898505 T^{2} + 40060 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 - 87296 T + 5453356606 T^{2} - 87296 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 70290 T + 5306812435 T^{2} - 70290 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 92580 T + 9976491505 T^{2} - 92580 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 72810 T + 5926578523 T^{2} + 72810 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 126140 T + 13936294470 T^{2} - 126140 p^{5} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$