# Properties

 Label 2-21e2-1.1-c3-0-29 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 $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 5·2-s + 17·4-s + 45·8-s + 68·11-s + 89·16-s + 340·22-s + 40·23-s − 125·25-s + 166·29-s + 85·32-s + 450·37-s − 180·43-s + 1.15e3·44-s + 200·46-s − 625·50-s − 590·53-s + 830·58-s − 287·64-s − 740·67-s − 688·71-s + 2.25e3·74-s − 1.38e3·79-s − 900·86-s + 3.06e3·88-s + 680·92-s − 2.12e3·100-s − 2.95e3·106-s + ⋯
 L(s)  = 1 + 1.76·2-s + 17/8·4-s + 1.98·8-s + 1.86·11-s + 1.39·16-s + 3.29·22-s + 0.362·23-s − 25-s + 1.06·29-s + 0.469·32-s + 1.99·37-s − 0.638·43-s + 3.96·44-s + 0.641·46-s − 1.76·50-s − 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s − 1.15·71-s + 3.53·74-s − 1.97·79-s − 1.12·86-s + 3.70·88-s + 0.770·92-s − 2.12·100-s − 2.70·106-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: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$6.197291470$$ $$L(\frac12)$$ $$\approx$$ $$6.197291470$$ $$L(\frac{5}{2})$$ 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 - 5 T + p^{3} T^{2}$$
5 $$1 + p^{3} T^{2}$$
11 $$1 - 68 T + p^{3} T^{2}$$
13 $$1 + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + p^{3} T^{2}$$
23 $$1 - 40 T + p^{3} T^{2}$$
29 $$1 - 166 T + p^{3} T^{2}$$
31 $$1 + p^{3} T^{2}$$
37 $$1 - 450 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 180 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + 590 T + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 + p^{3} T^{2}$$
67 $$1 + 740 T + p^{3} T^{2}$$
71 $$1 + 688 T + p^{3} T^{2}$$
73 $$1 + p^{3} T^{2}$$
79 $$1 + 1384 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$