# Properties

 Label 4-21e2-1.1-c7e2-0-2 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $43.0347$ Root an. cond. $2.56126$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9·2-s − 54·3-s + 71·4-s − 360·5-s + 486·6-s + 686·7-s − 1.70e3·8-s + 2.18e3·9-s + 3.24e3·10-s − 4.93e3·11-s − 3.83e3·12-s + 7.70e3·13-s − 6.17e3·14-s + 1.94e4·15-s + 8.58e3·16-s − 2.85e4·17-s − 1.96e4·18-s − 6.37e4·19-s − 2.55e4·20-s − 3.70e4·21-s + 4.43e4·22-s + 8.22e4·23-s + 9.18e4·24-s + 4.74e4·25-s − 6.93e4·26-s − 7.87e4·27-s + 4.87e4·28-s + ⋯
 L(s)  = 1 − 0.795·2-s − 1.15·3-s + 0.554·4-s − 1.28·5-s + 0.918·6-s + 0.755·7-s − 1.17·8-s + 9-s + 1.02·10-s − 1.11·11-s − 0.640·12-s + 0.973·13-s − 0.601·14-s + 1.48·15-s + 0.523·16-s − 1.41·17-s − 0.795·18-s − 2.13·19-s − 0.714·20-s − 0.872·21-s + 0.888·22-s + 1.40·23-s + 1.35·24-s + 0.607·25-s − 0.774·26-s − 0.769·27-s + 0.419·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$43.0347$$ Root analytic conductor: $$2.56126$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 441,\ (\ :7/2, 7/2),\ 1)$$

## Particular Values

 $$L(4)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{9}{2})$$ 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)$
bad3$C_1$ $$( 1 + p^{3} T )^{2}$$
7$C_1$ $$( 1 - p^{3} T )^{2}$$
good2$D_{4}$ $$1 + 9 T + 5 p T^{2} + 9 p^{7} T^{3} + p^{14} T^{4}$$
5$D_{4}$ $$1 + 72 p T + 3286 p^{2} T^{2} + 72 p^{8} T^{3} + p^{14} T^{4}$$
11$D_{4}$ $$1 + 4932 T + 19797958 T^{2} + 4932 p^{7} T^{3} + p^{14} T^{4}$$
13$D_{4}$ $$1 - 7708 T + 118266510 T^{2} - 7708 p^{7} T^{3} + p^{14} T^{4}$$
17$D_{4}$ $$1 + 28584 T + 942631150 T^{2} + 28584 p^{7} T^{3} + p^{14} T^{4}$$
19$D_{4}$ $$1 + 63728 T + 2569797414 T^{2} + 63728 p^{7} T^{3} + p^{14} T^{4}$$
23$D_{4}$ $$1 - 82260 T + 8202169294 T^{2} - 82260 p^{7} T^{3} + p^{14} T^{4}$$
29$D_{4}$ $$1 + 435996 T + 80862098782 T^{2} + 435996 p^{7} T^{3} + p^{14} T^{4}$$
31$D_{4}$ $$1 + 29240 T + 27798421182 T^{2} + 29240 p^{7} T^{3} + p^{14} T^{4}$$
37$D_{4}$ $$1 + 709556 T + 313296440190 T^{2} + 709556 p^{7} T^{3} + p^{14} T^{4}$$
41$D_{4}$ $$1 + 25056 T - 26592839954 T^{2} + 25056 p^{7} T^{3} + p^{14} T^{4}$$
43$D_{4}$ $$1 - 496216 T + 567160908438 T^{2} - 496216 p^{7} T^{3} + p^{14} T^{4}$$
47$D_{4}$ $$1 + 1575000 T + 1564490669086 T^{2} + 1575000 p^{7} T^{3} + p^{14} T^{4}$$
53$D_{4}$ $$1 - 2057436 T + 3149566808638 T^{2} - 2057436 p^{7} T^{3} + p^{14} T^{4}$$
59$D_{4}$ $$1 + 1101024 T + 4521593481622 T^{2} + 1101024 p^{7} T^{3} + p^{14} T^{4}$$
61$D_{4}$ $$1 - 28996 T + 5887677435486 T^{2} - 28996 p^{7} T^{3} + p^{14} T^{4}$$
67$D_{4}$ $$1 + 4480784 T + 16777543143750 T^{2} + 4480784 p^{7} T^{3} + p^{14} T^{4}$$
71$D_{4}$ $$1 - 54540 T + 17803030672942 T^{2} - 54540 p^{7} T^{3} + p^{14} T^{4}$$
73$D_{4}$ $$1 - 666604 T - 310686963642 T^{2} - 666604 p^{7} T^{3} + p^{14} T^{4}$$
79$D_{4}$ $$1 - 2322952 T + 38986658128734 T^{2} - 2322952 p^{7} T^{3} + p^{14} T^{4}$$
83$D_{4}$ $$1 + 7384392 T + 61089776413510 T^{2} + 7384392 p^{7} T^{3} + p^{14} T^{4}$$
89$D_{4}$ $$1 - 1784448 T + 26626978018894 T^{2} - 1784448 p^{7} T^{3} + p^{14} T^{4}$$
97$D_{4}$ $$1 - 16266412 T + 223770781220502 T^{2} - 16266412 p^{7} T^{3} + p^{14} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$