# Properties

 Degree $4$ Conductor $5531904$ Sign $1$ Motivic weight $3$ Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s − 12·5-s + 27·9-s + 4·11-s − 48·13-s + 72·15-s − 84·17-s − 72·19-s + 308·23-s − 44·25-s − 108·27-s − 80·29-s + 384·31-s − 24·33-s + 536·37-s + 288·39-s − 756·41-s − 400·43-s − 324·45-s + 312·47-s + 504·51-s − 52·53-s − 48·55-s + 432·57-s + 864·59-s − 1.41e3·61-s + 576·65-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1.07·5-s + 9-s + 0.109·11-s − 1.02·13-s + 1.23·15-s − 1.19·17-s − 0.869·19-s + 2.79·23-s − 0.351·25-s − 0.769·27-s − 0.512·29-s + 2.22·31-s − 0.126·33-s + 2.38·37-s + 1.18·39-s − 2.87·41-s − 1.41·43-s − 1.07·45-s + 0.968·47-s + 1.38·51-s − 0.134·53-s − 0.117·55-s + 1.00·57-s + 1.90·59-s − 2.97·61-s + 1.09·65-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: $$2$$ Selberg data: $$(4,\ 5531904,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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$D_{4}$ $$1 + 12 T + 188 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 4 T - 862 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 48 T + 4088 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 84 T + 676 p T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 72 T + 14622 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 308 T + 44522 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 80 T + 18626 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 384 T + 92918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 536 T + 159018 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 756 T + 278276 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 400 T + 142566 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 312 T + 222182 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 52 T + 241982 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 864 T + 596990 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 1416 T + 938664 T^{2} + 1416 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 144 T + 550262 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 1524 T + 1208266 T^{2} - 1524 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 744 T + 241296 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 976 T + 532734 T^{2} + 976 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 108 T + 1330436 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 744 T + 432480 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$