Properties

 Label 2-21e2-21.20-c3-0-9 Degree $2$ Conductor $441$ Sign $0.970 + 0.239i$ 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 − 4.86i·2-s − 15.6·4-s + 12.7·5-s + 37.3i·8-s − 62.1i·10-s + 54.1i·11-s − 8.85i·13-s + 56.2·16-s − 68.9·17-s + 163. i·19-s − 200.·20-s + 263.·22-s + 93.9i·23-s + 37.9·25-s − 43.0·26-s + ⋯
 L(s)  = 1 − 1.72i·2-s − 1.95·4-s + 1.14·5-s + 1.64i·8-s − 1.96i·10-s + 1.48i·11-s − 0.188i·13-s + 0.878·16-s − 0.983·17-s + 1.97i·19-s − 2.23·20-s + 2.55·22-s + 0.851i·23-s + 0.303·25-s − 0.324·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.970 + 0.239i$ Analytic conductor: $$26.0198$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{441} (440, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :3/2),\ 0.970 + 0.239i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.410882887$$ $$L(\frac12)$$ $$\approx$$ $$1.410882887$$ $$L(\frac{5}{2})$$ 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 + 4.86iT - 8T^{2}$$
5 $$1 - 12.7T + 125T^{2}$$
11 $$1 - 54.1iT - 1.33e3T^{2}$$
13 $$1 + 8.85iT - 2.19e3T^{2}$$
17 $$1 + 68.9T + 4.91e3T^{2}$$
19 $$1 - 163. iT - 6.85e3T^{2}$$
23 $$1 - 93.9iT - 1.21e4T^{2}$$
29 $$1 - 119. iT - 2.43e4T^{2}$$
31 $$1 - 98.8iT - 2.97e4T^{2}$$
37 $$1 - 94.1T + 5.06e4T^{2}$$
41 $$1 + 259.T + 6.89e4T^{2}$$
43 $$1 - 5.01T + 7.95e4T^{2}$$
47 $$1 - 57.3T + 1.03e5T^{2}$$
53 $$1 + 470. iT - 1.48e5T^{2}$$
59 $$1 + 225.T + 2.05e5T^{2}$$
61 $$1 + 427. iT - 2.26e5T^{2}$$
67 $$1 - 163.T + 3.00e5T^{2}$$
71 $$1 - 79.8iT - 3.57e5T^{2}$$
73 $$1 - 769. iT - 3.89e5T^{2}$$
79 $$1 - 534.T + 4.93e5T^{2}$$
83 $$1 + 438.T + 5.71e5T^{2}$$
89 $$1 - 25.6T + 7.04e5T^{2}$$
97 $$1 + 1.38e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$