# Properties

 Label 2-21e2-1.1-c3-0-23 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.59·2-s − 5.44·4-s − 18.2·5-s + 21.4·8-s + 29.2·10-s + 61.2·11-s − 32.4·13-s + 9.23·16-s + 81.3·17-s + 20.9·19-s + 99.6·20-s − 97.9·22-s − 33.7·23-s + 209.·25-s + 51.8·26-s + 52.0·29-s − 193.·31-s − 186.·32-s − 129.·34-s − 267.·37-s − 33.4·38-s − 393.·40-s − 203.·41-s − 21.9·43-s − 333.·44-s + 53.9·46-s + 247.·47-s + ⋯
 L(s)  = 1 − 0.564·2-s − 0.680·4-s − 1.63·5-s + 0.949·8-s + 0.924·10-s + 1.67·11-s − 0.692·13-s + 0.144·16-s + 1.16·17-s + 0.252·19-s + 1.11·20-s − 0.948·22-s − 0.305·23-s + 1.67·25-s + 0.391·26-s + 0.333·29-s − 1.12·31-s − 1.03·32-s − 0.655·34-s − 1.18·37-s − 0.142·38-s − 1.55·40-s − 0.773·41-s − 0.0778·43-s − 1.14·44-s + 0.172·46-s + 0.769·47-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: $$1$$ 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 + 1.59T + 8T^{2}$$
5 $$1 + 18.2T + 125T^{2}$$
11 $$1 - 61.2T + 1.33e3T^{2}$$
13 $$1 + 32.4T + 2.19e3T^{2}$$
17 $$1 - 81.3T + 4.91e3T^{2}$$
19 $$1 - 20.9T + 6.85e3T^{2}$$
23 $$1 + 33.7T + 1.21e4T^{2}$$
29 $$1 - 52.0T + 2.43e4T^{2}$$
31 $$1 + 193.T + 2.97e4T^{2}$$
37 $$1 + 267.T + 5.06e4T^{2}$$
41 $$1 + 203.T + 6.89e4T^{2}$$
43 $$1 + 21.9T + 7.95e4T^{2}$$
47 $$1 - 247.T + 1.03e5T^{2}$$
53 $$1 - 140.T + 1.48e5T^{2}$$
59 $$1 - 221.T + 2.05e5T^{2}$$
61 $$1 + 652.T + 2.26e5T^{2}$$
67 $$1 - 604.T + 3.00e5T^{2}$$
71 $$1 - 716.T + 3.57e5T^{2}$$
73 $$1 + 388.T + 3.89e5T^{2}$$
79 $$1 + 289.T + 4.93e5T^{2}$$
83 $$1 - 115.T + 5.71e5T^{2}$$
89 $$1 + 939.T + 7.04e5T^{2}$$
97 $$1 + 120.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

−10.11475228142303876399599195621, −9.227059925730514287466817924483, −8.456655412017598751789836830545, −7.62831977428277707497391886396, −6.91444832273275243292503653981, −5.25078624997037342326830925106, −4.12511813104430031466636897007, −3.52696004030330274073052593649, −1.23583670498177209247174643099, 0, 1.23583670498177209247174643099, 3.52696004030330274073052593649, 4.12511813104430031466636897007, 5.25078624997037342326830925106, 6.91444832273275243292503653981, 7.62831977428277707497391886396, 8.456655412017598751789836830545, 9.227059925730514287466817924483, 10.11475228142303876399599195621