# Properties

 Label 2-21e2-1.1-c3-0-32 Degree $2$ Conductor $441$ Sign $-1$ Analytic cond. $26.0198$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 8·4-s + 70·13-s + 64·16-s − 56·19-s − 125·25-s − 308·31-s + 110·37-s − 520·43-s − 560·52-s − 182·61-s − 512·64-s − 880·67-s − 1.19e3·73-s + 448·76-s + 884·79-s + 1.33e3·97-s + 1.00e3·100-s − 1.82e3·103-s − 646·109-s + ⋯
 L(s)  = 1 − 4-s + 1.49·13-s + 16-s − 0.676·19-s − 25-s − 1.78·31-s + 0.488·37-s − 1.84·43-s − 1.49·52-s − 0.382·61-s − 64-s − 1.60·67-s − 1.90·73-s + 0.676·76-s + 1.25·79-s + 1.39·97-s + 100-s − 1.74·103-s − 0.567·109-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{441} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + p^{3} T^{2}$$
5 $$1 + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 - 70 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + 56 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 + 308 T + p^{3} T^{2}$$
37 $$1 - 110 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 520 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 + 182 T + p^{3} T^{2}$$
67 $$1 + 880 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 + 1190 T + p^{3} T^{2}$$
79 $$1 - 884 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 - 1330 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$