# Properties

 Label 2-21e2-1.1-c3-0-24 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 − 5.30·2-s + 20.1·4-s − 5.56·5-s − 64.6·8-s + 29.5·10-s + 13.9·11-s + 38.6·13-s + 181.·16-s − 43.4·17-s − 109.·19-s − 112.·20-s − 73.8·22-s + 74.8·23-s − 94.0·25-s − 205.·26-s + 72.3·29-s + 64.0·31-s − 447.·32-s + 230.·34-s + 188.·37-s + 578.·38-s + 359.·40-s + 24.7·41-s − 243.·43-s + 280.·44-s − 397.·46-s + 620.·47-s + ⋯
 L(s)  = 1 − 1.87·2-s + 2.52·4-s − 0.497·5-s − 2.85·8-s + 0.933·10-s + 0.381·11-s + 0.825·13-s + 2.83·16-s − 0.620·17-s − 1.31·19-s − 1.25·20-s − 0.715·22-s + 0.678·23-s − 0.752·25-s − 1.54·26-s + 0.463·29-s + 0.371·31-s − 2.47·32-s + 1.16·34-s + 0.838·37-s + 2.47·38-s + 1.42·40-s + 0.0944·41-s − 0.864·43-s + 0.962·44-s − 1.27·46-s + 1.92·47-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 + 5.30T + 8T^{2}$$
5 $$1 + 5.56T + 125T^{2}$$
11 $$1 - 13.9T + 1.33e3T^{2}$$
13 $$1 - 38.6T + 2.19e3T^{2}$$
17 $$1 + 43.4T + 4.91e3T^{2}$$
19 $$1 + 109.T + 6.85e3T^{2}$$
23 $$1 - 74.8T + 1.21e4T^{2}$$
29 $$1 - 72.3T + 2.43e4T^{2}$$
31 $$1 - 64.0T + 2.97e4T^{2}$$
37 $$1 - 188.T + 5.06e4T^{2}$$
41 $$1 - 24.7T + 6.89e4T^{2}$$
43 $$1 + 243.T + 7.95e4T^{2}$$
47 $$1 - 620.T + 1.03e5T^{2}$$
53 $$1 - 287.T + 1.48e5T^{2}$$
59 $$1 + 525.T + 2.05e5T^{2}$$
61 $$1 + 383.T + 2.26e5T^{2}$$
67 $$1 - 198.T + 3.00e5T^{2}$$
71 $$1 + 785.T + 3.57e5T^{2}$$
73 $$1 + 331.T + 3.89e5T^{2}$$
79 $$1 - 437.T + 4.93e5T^{2}$$
83 $$1 + 241.T + 5.71e5T^{2}$$
89 $$1 + 1.58e3T + 7.04e5T^{2}$$
97 $$1 - 79.2T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$