# Properties

 Label 2-21e2-21.20-c3-0-22 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 $0$

# Learn more about

## Dirichlet series

 L(s)  = 1 − 0.748i·2-s + 7.43·4-s + 10.8·5-s − 11.5i·8-s − 8.13i·10-s + 51.8i·11-s + 32.1i·13-s + 50.8·16-s + 81.4·17-s − 0.0485i·19-s + 80.7·20-s + 38.8·22-s + 89.2i·23-s − 7.16·25-s + 24.1·26-s + ⋯
 L(s)  = 1 − 0.264i·2-s + 0.929·4-s + 0.970·5-s − 0.511i·8-s − 0.257i·10-s + 1.42i·11-s + 0.686i·13-s + 0.794·16-s + 1.16·17-s − 0.000586i·19-s + 0.902·20-s + 0.376·22-s + 0.809i·23-s − 0.0572·25-s + 0.181·26-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

 $$L(2)$$ $$\approx$$ $$3.020531106$$ $$L(\frac12)$$ $$\approx$$ $$3.020531106$$ $$L(\frac{5}{2})$$ 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 + 0.748iT - 8T^{2}$$
5 $$1 - 10.8T + 125T^{2}$$
11 $$1 - 51.8iT - 1.33e3T^{2}$$
13 $$1 - 32.1iT - 2.19e3T^{2}$$
17 $$1 - 81.4T + 4.91e3T^{2}$$
19 $$1 + 0.0485iT - 6.85e3T^{2}$$
23 $$1 - 89.2iT - 1.21e4T^{2}$$
29 $$1 - 175. iT - 2.43e4T^{2}$$
31 $$1 + 215. iT - 2.97e4T^{2}$$
37 $$1 - 64.5T + 5.06e4T^{2}$$
41 $$1 + 411.T + 6.89e4T^{2}$$
43 $$1 + 234.T + 7.95e4T^{2}$$
47 $$1 - 632.T + 1.03e5T^{2}$$
53 $$1 - 265. iT - 1.48e5T^{2}$$
59 $$1 - 351.T + 2.05e5T^{2}$$
61 $$1 + 778. iT - 2.26e5T^{2}$$
67 $$1 + 196.T + 3.00e5T^{2}$$
71 $$1 + 142. iT - 3.57e5T^{2}$$
73 $$1 + 780. iT - 3.89e5T^{2}$$
79 $$1 - 1.28e3T + 4.93e5T^{2}$$
83 $$1 - 235.T + 5.71e5T^{2}$$
89 $$1 + 670.T + 7.04e5T^{2}$$
97 $$1 + 655. iT - 9.12e5T^{2}$$
show more
show less
$$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

−10.60600960819911199574221792430, −9.891030161176818162027386260019, −9.315840825090139994845979772011, −7.75900157725666868485667894750, −7.01080150064487969525590108289, −6.08402155143305887655457977358, −5.08909793914144832362270465596, −3.61351536217794674647919818685, −2.23812752142122884307037754602, −1.50399390311150455003408642091, 1.03700922907362904621671928705, 2.44605913056545081633274440601, 3.43562033796311900173312378013, 5.38795214908183952313007967346, 5.88826699371342901423450742720, 6.78582008971462343752649251545, 7.951785806413539789406984631278, 8.679799954411843181314188860464, 10.05580688459286736287321403919, 10.50111807200530857330116419053