# Properties

 Label 2-3e6-81.67-c1-0-18 Degree $2$ Conductor $729$ Sign $0.907 - 0.419i$ Analytic cond. $5.82109$ Root an. cond. $2.41269$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.652 + 0.429i)2-s + (−0.550 + 1.27i)4-s + (−1.01 + 1.07i)5-s + (3.77 + 0.441i)7-s + (−0.459 − 2.60i)8-s + (0.200 − 1.13i)10-s + (1.44 − 4.84i)11-s + (0.261 − 4.48i)13-s + (−2.65 + 1.33i)14-s + (−0.485 − 0.515i)16-s + (4.30 − 1.56i)17-s + (4.19 + 1.52i)19-s + (−0.814 − 1.88i)20-s + (1.13 + 3.78i)22-s + (−3.43 + 0.401i)23-s + ⋯
 L(s)  = 1 + (−0.461 + 0.303i)2-s + (−0.275 + 0.637i)4-s + (−0.454 + 0.481i)5-s + (1.42 + 0.166i)7-s + (−0.162 − 0.922i)8-s + (0.0635 − 0.360i)10-s + (0.437 − 1.45i)11-s + (0.0724 − 1.24i)13-s + (−0.709 + 0.356i)14-s + (−0.121 − 0.128i)16-s + (1.04 − 0.380i)17-s + (0.962 + 0.350i)19-s + (−0.182 − 0.422i)20-s + (0.241 + 0.806i)22-s + (−0.715 + 0.0836i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $0.907 - 0.419i$ Analytic conductor: $$5.82109$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{729} (28, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ 0.907 - 0.419i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.20034 + 0.263926i$$ $$L(\frac12)$$ $$\approx$$ $$1.20034 + 0.263926i$$ $$L(\frac{3}{2})$$ 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 + (0.652 - 0.429i)T + (0.792 - 1.83i)T^{2}$$
5 $$1 + (1.01 - 1.07i)T + (-0.290 - 4.99i)T^{2}$$
7 $$1 + (-3.77 - 0.441i)T + (6.81 + 1.61i)T^{2}$$
11 $$1 + (-1.44 + 4.84i)T + (-9.19 - 6.04i)T^{2}$$
13 $$1 + (-0.261 + 4.48i)T + (-12.9 - 1.50i)T^{2}$$
17 $$1 + (-4.30 + 1.56i)T + (13.0 - 10.9i)T^{2}$$
19 $$1 + (-4.19 - 1.52i)T + (14.5 + 12.2i)T^{2}$$
23 $$1 + (3.43 - 0.401i)T + (22.3 - 5.30i)T^{2}$$
29 $$1 + (0.583 + 0.293i)T + (17.3 + 23.2i)T^{2}$$
31 $$1 + (-0.393 - 0.527i)T + (-8.89 + 29.6i)T^{2}$$
37 $$1 + (-0.766 - 0.642i)T + (6.42 + 36.4i)T^{2}$$
41 $$1 + (0.570 + 0.375i)T + (16.2 + 37.6i)T^{2}$$
43 $$1 + (-8.16 + 1.93i)T + (38.4 - 19.2i)T^{2}$$
47 $$1 + (4.73 - 6.36i)T + (-13.4 - 45.0i)T^{2}$$
53 $$1 + (-2.07 + 3.59i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-1.51 - 5.04i)T + (-49.2 + 32.4i)T^{2}$$
61 $$1 + (2.68 + 6.23i)T + (-41.8 + 44.3i)T^{2}$$
67 $$1 + (-4.73 + 2.37i)T + (40.0 - 53.7i)T^{2}$$
71 $$1 + (-1.06 + 6.03i)T + (-66.7 - 24.2i)T^{2}$$
73 $$1 + (0.764 + 4.33i)T + (-68.5 + 24.9i)T^{2}$$
79 $$1 + (9.39 - 6.17i)T + (31.2 - 72.5i)T^{2}$$
83 $$1 + (-1.35 + 0.891i)T + (32.8 - 76.2i)T^{2}$$
89 $$1 + (-0.181 - 1.02i)T + (-83.6 + 30.4i)T^{2}$$
97 $$1 + (3.42 + 3.62i)T + (-5.64 + 96.8i)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

−10.48391691597502968573179074243, −9.419618605637260429359507348572, −8.451220265930111974989945309991, −7.86644576340617017355862234944, −7.46436700617226333459417147486, −6.00928257286495375775346945993, −5.13619792979055756906567733685, −3.73972358904872291090422091235, −3.06663216195434977388881801622, −0.974672334507125158497478640242, 1.22628033155649267384805236127, 2.04561547723065660224131106630, 4.21402829247815230209878881343, 4.69043834269134934910647644455, 5.68098354053661539832290933810, 7.06316495254274509140686462000, 7.923244029918822556919934418624, 8.683442040499821607908605478964, 9.569943201725254874279293480736, 10.16845722524255793535002421540