# 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 − 12·5-s + 27·9-s − 4·11-s − 48·13-s − 72·15-s − 132·17-s + 120·19-s + 76·23-s − 140·25-s + 108·27-s − 112·29-s + 432·31-s − 24·33-s − 280·37-s − 288·39-s − 36·41-s + 128·43-s − 324·45-s − 264·47-s − 792·51-s + 268·53-s + 48·55-s + 720·57-s + 336·59-s + 504·61-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 − 1.88·17-s + 1.44·19-s + 0.689·23-s − 1.11·25-s + 0.769·27-s − 0.717·29-s + 2.50·31-s − 0.126·33-s − 1.24·37-s − 1.18·39-s − 0.137·41-s + 0.453·43-s − 1.07·45-s − 0.819·47-s − 2.17·51-s + 0.694·53-s + 0.117·55-s + 1.67·57-s + 0.741·59-s + 1.05·61-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: $$0$$ Selberg data: $$(4,\ 5531904,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$4.207991965$$ $$L(\frac12)$$ $$\approx$$ $$4.207991965$$ $$L(\frac{5}{2})$$ 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$D_{4}$ $$1 + 12 T + 284 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 4 T + 2594 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 48 T + 4520 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 132 T + 820 p T^{2} + 132 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 120 T + 13086 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 76 T + 4970 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 112 T + 6914 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 432 T + 100406 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 280 T + 56106 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 36 T + 105908 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 128 T - 2778 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 264 T - 3418 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 268 T + 241982 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 336 T + 261374 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 - 504 T + 268248 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 384 T + 596918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 396 T + 521098 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 312 T + 94320 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 848 T + 985854 T^{2} - 848 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 648 T + 1235750 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 612 T + 1442324 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 2184 T + 2982432 T^{2} - 2184 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$