# Properties

 Label 4-21e2-1.1-c3e2-0-0 Degree $4$ Conductor $441$ Sign $1$ Analytic cond. $1.53522$ Root an. cond. $1.11312$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 6·3-s + 5·4-s + 6·5-s − 18·6-s + 14·7-s − 27·8-s + 27·9-s − 18·10-s − 6·11-s + 30·12-s + 16·13-s − 42·14-s + 36·15-s + 69·16-s − 6·17-s − 81·18-s + 64·19-s + 30·20-s + 84·21-s + 18·22-s + 6·23-s − 162·24-s − 166·25-s − 48·26-s + 108·27-s + 70·28-s + ⋯
 L(s)  = 1 − 1.06·2-s + 1.15·3-s + 5/8·4-s + 0.536·5-s − 1.22·6-s + 0.755·7-s − 1.19·8-s + 9-s − 0.569·10-s − 0.164·11-s + 0.721·12-s + 0.341·13-s − 0.801·14-s + 0.619·15-s + 1.07·16-s − 0.0856·17-s − 1.06·18-s + 0.772·19-s + 0.335·20-s + 0.872·21-s + 0.174·22-s + 0.0543·23-s − 1.37·24-s − 1.32·25-s − 0.362·26-s + 0.769·27-s + 0.472·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$1.53522$$ Root analytic conductor: $$1.11312$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{21} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 441,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.057700512$$ $$L(\frac12)$$ $$\approx$$ $$1.057700512$$ $$L(\frac{5}{2})$$ 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)$
bad3$C_1$ $$( 1 - p T )^{2}$$
7$C_1$ $$( 1 - p T )^{2}$$
good2$D_{4}$ $$1 + 3 T + p^{2} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 - 6 T + 202 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 6 T + 1246 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 - 16 T + 2406 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 64 T + 6534 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 6 T + 7870 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 252 T + 56446 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 - 40 T - 13890 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 1104 T + 602230 T^{2} + 1104 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 428 T + 425886 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 - 954 T + 13106 p T^{2} - 954 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 1072 T + 1063278 T^{2} - 1072 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 - 808 T + 903054 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$