# Properties

 Label 2-3e6-27.13-c1-0-17 Degree $2$ Conductor $729$ Sign $0.893 + 0.448i$ Analytic cond. $5.82109$ Root an. cond. $2.41269$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.152 − 0.866i)2-s + (1.15 − 0.419i)4-s + (2.97 + 2.49i)5-s + (2.05 + 0.747i)7-s + (−1.41 − 2.45i)8-s + (1.70 − 2.95i)10-s + (0.124 − 0.104i)11-s + (0.418 − 2.37i)13-s + (0.333 − 1.89i)14-s + (−0.0320 + 0.0269i)16-s + (−1.5 + 2.59i)17-s + (−1.79 − 3.11i)19-s + (4.47 + 1.62i)20-s + (−0.109 − 0.0918i)22-s + (−2.66 + 0.970i)23-s + ⋯
 L(s)  = 1 + (−0.107 − 0.612i)2-s + (0.576 − 0.209i)4-s + (1.32 + 1.11i)5-s + (0.775 + 0.282i)7-s + (−0.501 − 0.868i)8-s + (0.539 − 0.934i)10-s + (0.0375 − 0.0314i)11-s + (0.116 − 0.658i)13-s + (0.0891 − 0.505i)14-s + (−0.00802 + 0.00673i)16-s + (−0.363 + 0.630i)17-s + (−0.412 − 0.714i)19-s + (0.999 + 0.363i)20-s + (−0.0233 − 0.0195i)22-s + (−0.555 + 0.202i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $0.893 + 0.448i$ Analytic conductor: $$5.82109$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{729} (82, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ 0.893 + 0.448i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.16986 - 0.514267i$$ $$L(\frac12)$$ $$\approx$$ $$2.16986 - 0.514267i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + (0.152 + 0.866i)T + (-1.87 + 0.684i)T^{2}$$
5 $$1 + (-2.97 - 2.49i)T + (0.868 + 4.92i)T^{2}$$
7 $$1 + (-2.05 - 0.747i)T + (5.36 + 4.49i)T^{2}$$
11 $$1 + (-0.124 + 0.104i)T + (1.91 - 10.8i)T^{2}$$
13 $$1 + (-0.418 + 2.37i)T + (-12.2 - 4.44i)T^{2}$$
17 $$1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.79 + 3.11i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (2.66 - 0.970i)T + (17.6 - 14.7i)T^{2}$$
29 $$1 + (-1.16 - 6.61i)T + (-27.2 + 9.91i)T^{2}$$
31 $$1 + (-4.87 + 1.77i)T + (23.7 - 19.9i)T^{2}$$
37 $$1 + (-3.31 + 5.74i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (1.00 - 5.71i)T + (-38.5 - 14.0i)T^{2}$$
43 $$1 + (4.76 - 4.00i)T + (7.46 - 42.3i)T^{2}$$
47 $$1 + (6.95 + 2.52i)T + (36.0 + 30.2i)T^{2}$$
53 $$1 - 1.40T + 53T^{2}$$
59 $$1 + (3.92 + 3.29i)T + (10.2 + 58.1i)T^{2}$$
61 $$1 + (-3.55 - 1.29i)T + (46.7 + 39.2i)T^{2}$$
67 $$1 + (1.01 - 5.77i)T + (-62.9 - 22.9i)T^{2}$$
71 $$1 + (-7.65 + 13.2i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (4.34 + 7.51i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (0.220 + 1.24i)T + (-74.2 + 27.0i)T^{2}$$
83 $$1 + (1.47 + 8.34i)T + (-77.9 + 28.3i)T^{2}$$
89 $$1 + (3.86 + 6.68i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (2.99 - 2.51i)T + (16.8 - 95.5i)T^{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

−10.39697116324914486139381562293, −9.844328765993893990547403473470, −8.834949151580960449205079709952, −7.66556837023332507710448939965, −6.53429821077039435060168156260, −6.12956871411968940346350453109, −5.01838269263494668593048493759, −3.32649726448077348653065548572, −2.41541886104969997472809390485, −1.60305492429182375595316581281, 1.51030829205420779291072382727, 2.41926274959492276510373789580, 4.33162500712435502897221860025, 5.21086285817132883469804942898, 6.09697823734135842233130718926, 6.79415860196811787836838872731, 8.080202345705499031937059156641, 8.500775724321143286642355564143, 9.520045017776198272265185940876, 10.27135025806276971277826817256