# Properties

 Degree $4$ Conductor $5531904$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s − 14·5-s + 27·9-s − 18·11-s − 48·13-s − 84·15-s − 34·17-s − 16·19-s − 110·23-s + 74·25-s + 108·27-s + 212·29-s − 136·31-s − 108·33-s − 24·37-s − 288·39-s − 694·41-s + 584·43-s − 378·45-s − 316·47-s − 204·51-s + 560·53-s + 252·55-s − 96·57-s − 492·59-s + 604·61-s + 672·65-s + ⋯
 L(s)  = 1 + 1.15·3-s − 1.25·5-s + 9-s − 0.493·11-s − 1.02·13-s − 1.44·15-s − 0.485·17-s − 0.193·19-s − 0.997·23-s + 0.591·25-s + 0.769·27-s + 1.35·29-s − 0.787·31-s − 0.569·33-s − 0.106·37-s − 1.18·39-s − 2.64·41-s + 2.07·43-s − 1.25·45-s − 0.980·47-s − 0.560·51-s + 1.45·53-s + 0.617·55-s − 0.223·57-s − 1.08·59-s + 1.26·61-s + 1.28·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$5531904$$    =    $$2^{8} \cdot 3^{2} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$3$$ Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 5531904,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.370399596$$ $$L(\frac12)$$ $$\approx$$ $$3.370399596$$ $$L(\frac{5}{2})$$ 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 $$1$$
3$C_1$ $$( 1 - p T )^{2}$$
7 $$1$$
good5$D_{4}$ $$1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 18 T + 1150 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 48 T + 4262 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 2 p T - 4222 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 16 T + 10950 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 110 T + 18686 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 212 T + 57182 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 136 T + 61374 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 694 T + 243914 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 316 T + 231902 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 560 T + 369782 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 492 T + 351622 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 - 604 T + 406398 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 1710 T + 1336222 T^{2} - 1710 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 1312 T + 1201998 T^{2} - 1312 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 556 T + 751134 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 264 T + 979750 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 70 T - 186262 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 136 T + 1812270 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$