# Properties

 Label 2-21e2-7.2-c3-0-24 Degree $2$ Conductor $441$ Sign $0.991 + 0.126i$ Analytic cond. $26.0198$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.32 − 2.29i)2-s + (0.5 + 0.866i)4-s + 23.8·8-s + (13.2 + 22.9i)11-s + (27.5 − 47.6i)16-s + 70·22-s + (−108. + 187. i)23-s + (62.5 + 108. i)25-s + 264.·29-s + (22.4 + 38.9i)32-s + (225 − 389. i)37-s + 180·43-s + (−13.2 + 22.9i)44-s + (287 + 497. i)46-s + 330.·50-s + ⋯
 L(s)  = 1 + (0.467 − 0.810i)2-s + (0.0625 + 0.108i)4-s + 1.05·8-s + (0.362 + 0.628i)11-s + (0.429 − 0.744i)16-s + 0.678·22-s + (−0.983 + 1.70i)23-s + (0.5 + 0.866i)25-s + 1.69·29-s + (0.124 + 0.215i)32-s + (0.999 − 1.73i)37-s + 0.638·43-s + (−0.0453 + 0.0785i)44-s + (0.919 + 1.59i)46-s + 0.935·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.991 + 0.126i$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{441} (226, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 0.991 + 0.126i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.877145539$$ $$L(\frac12)$$ $$\approx$$ $$2.877145539$$ $$L(\frac{5}{2})$$ 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 + (-1.32 + 2.29i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-13.2 - 22.9i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 2.19e3T^{2}$$
17 $$1 + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (108. - 187. i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 264.T + 2.43e4T^{2}$$
31 $$1 + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-225 + 389. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 180T + 7.95e4T^{2}$$
47 $$1 + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-248. - 430. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-370 - 640. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 978.T + 3.57e5T^{2}$$
73 $$1 + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$