# Properties

 Label 2-21e2-1.1-c3-0-40 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 $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5.39·2-s + 21.1·4-s + 15.5·5-s + 71.0·8-s + 84.1·10-s + 31.9·11-s − 72.5·13-s + 214.·16-s − 29.0·17-s − 108.·19-s + 329.·20-s + 172.·22-s − 55.2·23-s + 117.·25-s − 391.·26-s + 17.7·29-s − 56.1·31-s + 589.·32-s − 156.·34-s − 295.·37-s − 587.·38-s + 1.10e3·40-s + 238.·41-s + 16.8·43-s + 676.·44-s − 298.·46-s + 511.·47-s + ⋯
 L(s)  = 1 + 1.90·2-s + 2.64·4-s + 1.39·5-s + 3.14·8-s + 2.65·10-s + 0.876·11-s − 1.54·13-s + 3.35·16-s − 0.414·17-s − 1.31·19-s + 3.68·20-s + 1.67·22-s − 0.501·23-s + 0.941·25-s − 2.95·26-s + 0.113·29-s − 0.325·31-s + 3.25·32-s − 0.790·34-s − 1.31·37-s − 2.50·38-s + 4.37·40-s + 0.908·41-s + 0.0596·43-s + 2.31·44-s − 0.956·46-s + 1.58·47-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

 $$L(2)$$ $$\approx$$ $$7.867223976$$ $$L(\frac12)$$ $$\approx$$ $$7.867223976$$ $$L(\frac{5}{2})$$ 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 - 5.39T + 8T^{2}$$
5 $$1 - 15.5T + 125T^{2}$$
11 $$1 - 31.9T + 1.33e3T^{2}$$
13 $$1 + 72.5T + 2.19e3T^{2}$$
17 $$1 + 29.0T + 4.91e3T^{2}$$
19 $$1 + 108.T + 6.85e3T^{2}$$
23 $$1 + 55.2T + 1.21e4T^{2}$$
29 $$1 - 17.7T + 2.43e4T^{2}$$
31 $$1 + 56.1T + 2.97e4T^{2}$$
37 $$1 + 295.T + 5.06e4T^{2}$$
41 $$1 - 238.T + 6.89e4T^{2}$$
43 $$1 - 16.8T + 7.95e4T^{2}$$
47 $$1 - 511.T + 1.03e5T^{2}$$
53 $$1 - 265.T + 1.48e5T^{2}$$
59 $$1 + 254.T + 2.05e5T^{2}$$
61 $$1 - 72.8T + 2.26e5T^{2}$$
67 $$1 + 506.T + 3.00e5T^{2}$$
71 $$1 + 827.T + 3.57e5T^{2}$$
73 $$1 - 372.T + 3.89e5T^{2}$$
79 $$1 - 1.02e3T + 4.93e5T^{2}$$
83 $$1 - 453.T + 5.71e5T^{2}$$
89 $$1 - 332.T + 7.04e5T^{2}$$
97 $$1 - 1.16e3T + 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

−10.86069240612887171751226808446, −10.14566986301809319416073516147, −9.049435105398708279942348994521, −7.36768464887901727385991048708, −6.52502956480527093864955048262, −5.84126734760816697635337494646, −4.91845599433010237738221723715, −4.01298064682546215002871575029, −2.54341338814675595787958961351, −1.87472276351988175545453895981, 1.87472276351988175545453895981, 2.54341338814675595787958961351, 4.01298064682546215002871575029, 4.91845599433010237738221723715, 5.84126734760816697635337494646, 6.52502956480527093864955048262, 7.36768464887901727385991048708, 9.049435105398708279942348994521, 10.14566986301809319416073516147, 10.86069240612887171751226808446