# Properties

 Label 2-21e2-1.1-c3-0-2 Degree $2$ Conductor $441$ Sign $1$ Analytic cond. $26.0198$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 0.248·2-s − 7.93·4-s − 12.4·5-s + 3.95·8-s + 3.08·10-s − 60.3·11-s − 36.4·13-s + 62.5·16-s − 48.7·17-s + 50.5·19-s + 98.7·20-s + 14.9·22-s − 138.·23-s + 29.6·25-s + 9.03·26-s + 61.1·29-s + 1.16·31-s − 47.1·32-s + 12.0·34-s + 69.5·37-s − 12.5·38-s − 49.1·40-s + 308.·41-s + 174.·43-s + 478.·44-s + 34.4·46-s + 389.·47-s + ⋯
 L(s)  = 1 − 0.0877·2-s − 0.992·4-s − 1.11·5-s + 0.174·8-s + 0.0975·10-s − 1.65·11-s − 0.777·13-s + 0.976·16-s − 0.695·17-s + 0.610·19-s + 1.10·20-s + 0.144·22-s − 1.25·23-s + 0.236·25-s + 0.0681·26-s + 0.391·29-s + 0.00677·31-s − 0.260·32-s + 0.0609·34-s + 0.308·37-s − 0.0535·38-s − 0.194·40-s + 1.17·41-s + 0.618·43-s + 1.64·44-s + 0.110·46-s + 1.20·47-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: no Arithmetic: yes Character: $\chi_{441} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.4740538218$$ $$L(\frac12)$$ $$\approx$$ $$0.4740538218$$ $$L(\frac{5}{2})$$ 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 + 0.248T + 8T^{2}$$
5 $$1 + 12.4T + 125T^{2}$$
11 $$1 + 60.3T + 1.33e3T^{2}$$
13 $$1 + 36.4T + 2.19e3T^{2}$$
17 $$1 + 48.7T + 4.91e3T^{2}$$
19 $$1 - 50.5T + 6.85e3T^{2}$$
23 $$1 + 138.T + 1.21e4T^{2}$$
29 $$1 - 61.1T + 2.43e4T^{2}$$
31 $$1 - 1.16T + 2.97e4T^{2}$$
37 $$1 - 69.5T + 5.06e4T^{2}$$
41 $$1 - 308.T + 6.89e4T^{2}$$
43 $$1 - 174.T + 7.95e4T^{2}$$
47 $$1 - 389.T + 1.03e5T^{2}$$
53 $$1 + 314.T + 1.48e5T^{2}$$
59 $$1 - 844.T + 2.05e5T^{2}$$
61 $$1 - 338.T + 2.26e5T^{2}$$
67 $$1 + 971.T + 3.00e5T^{2}$$
71 $$1 - 98.4T + 3.57e5T^{2}$$
73 $$1 + 710.T + 3.89e5T^{2}$$
79 $$1 + 486.T + 4.93e5T^{2}$$
83 $$1 - 605.T + 5.71e5T^{2}$$
89 $$1 - 218.T + 7.04e5T^{2}$$
97 $$1 - 782.T + 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.57461814882417870438243676543, −9.856750590316041731953299311717, −8.783582212684113623554611445132, −7.86662009277336480425881755708, −7.46652292662645158143523961697, −5.74686645541848713865118725596, −4.76216996535030756242696614831, −3.96165274207307934936517426013, −2.61159323199832957327599515567, −0.42722531252300184311001805753, 0.42722531252300184311001805753, 2.61159323199832957327599515567, 3.96165274207307934936517426013, 4.76216996535030756242696614831, 5.74686645541848713865118725596, 7.46652292662645158143523961697, 7.86662009277336480425881755708, 8.783582212684113623554611445132, 9.856750590316041731953299311717, 10.57461814882417870438243676543