# Properties

 Label 4-2352e2-1.1-c3e2-0-19 Degree $4$ Conductor $5531904$ Sign $1$ Analytic cond. $19257.8$ Root an. cond. $11.7801$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s − 11·5-s + 27·9-s + 5·11-s − 5·13-s − 66·15-s − 100·17-s + 67·19-s − 76·23-s − 111·25-s + 108·27-s + 275·29-s + 362·31-s + 30·33-s − 5·37-s − 30·39-s + 162·41-s − 721·43-s − 297·45-s − 216·47-s − 600·51-s + 495·53-s − 55·55-s + 402·57-s + 173·59-s + 532·61-s + 55·65-s + ⋯
 L(s)  = 1 + 1.15·3-s − 0.983·5-s + 9-s + 0.137·11-s − 0.106·13-s − 1.13·15-s − 1.42·17-s + 0.808·19-s − 0.689·23-s − 0.887·25-s + 0.769·27-s + 1.76·29-s + 2.09·31-s + 0.158·33-s − 0.0222·37-s − 0.123·39-s + 0.617·41-s − 2.55·43-s − 0.983·45-s − 0.670·47-s − 1.64·51-s + 1.28·53-s − 0.134·55-s + 0.934·57-s + 0.381·59-s + 1.11·61-s + 0.104·65-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: $1$ Analytic conductor: $$19257.8$$ Root analytic conductor: $$11.7801$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes 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$D_{4}$ $$1 + 11 T + 232 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 5 T + 304 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 5 T + 3194 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 100 T + 11554 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 67 T + 14406 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 76 T + 6478 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 275 T + 61846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 362 T + 73043 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 5 T + 3606 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 162 T + 128770 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 + 721 T + 288540 T^{2} + 721 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 216 T + 156778 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 - 495 T + 355102 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 173 T + 282706 T^{2} - 173 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 - 532 T + 447518 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 111 T + 318532 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 + 1215 T + 1011556 T^{2} + 1215 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 1460 T + 1333505 T^{2} + 1460 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 1409 T + 1147696 T^{2} + 1409 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 1974 T + 2298994 T^{2} - 1974 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 561 T + 1863448 T^{2} + 561 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$