# Properties

 Label 2-3e6-1.1-c1-0-18 Degree $2$ Conductor $729$ Sign $1$ Analytic cond. $5.82109$ Root an. cond. $2.41269$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2.40·2-s + 3.76·4-s − 0.0930·5-s − 0.579·7-s + 4.24·8-s − 0.223·10-s + 3.09·11-s + 4.20·13-s − 1.39·14-s + 2.66·16-s + 1.99·17-s − 3.84·19-s − 0.350·20-s + 7.42·22-s + 4.45·23-s − 4.99·25-s + 10.0·26-s − 2.18·28-s − 6.39·29-s + 1.65·31-s − 2.10·32-s + 4.78·34-s + 0.0539·35-s + 4.03·37-s − 9.23·38-s − 0.395·40-s + 1.09·41-s + ⋯
 L(s)  = 1 + 1.69·2-s + 1.88·4-s − 0.0416·5-s − 0.219·7-s + 1.50·8-s − 0.0706·10-s + 0.932·11-s + 1.16·13-s − 0.372·14-s + 0.665·16-s + 0.482·17-s − 0.882·19-s − 0.0784·20-s + 1.58·22-s + 0.928·23-s − 0.998·25-s + 1.97·26-s − 0.412·28-s − 1.18·29-s + 0.297·31-s − 0.371·32-s + 0.820·34-s + 0.00912·35-s + 0.662·37-s − 1.49·38-s − 0.0624·40-s + 0.171·41-s + ⋯

## Functional equation

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

## Invariants

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

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.032184741$$ $$L(\frac12)$$ $$\approx$$ $$4.032184741$$ $$L(\frac{3}{2})$$ 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 - 2.40T + 2T^{2}$$
5 $$1 + 0.0930T + 5T^{2}$$
7 $$1 + 0.579T + 7T^{2}$$
11 $$1 - 3.09T + 11T^{2}$$
13 $$1 - 4.20T + 13T^{2}$$
17 $$1 - 1.99T + 17T^{2}$$
19 $$1 + 3.84T + 19T^{2}$$
23 $$1 - 4.45T + 23T^{2}$$
29 $$1 + 6.39T + 29T^{2}$$
31 $$1 - 1.65T + 31T^{2}$$
37 $$1 - 4.03T + 37T^{2}$$
41 $$1 - 1.09T + 41T^{2}$$
43 $$1 + 6.90T + 43T^{2}$$
47 $$1 + 3.59T + 47T^{2}$$
53 $$1 + 5.40T + 53T^{2}$$
59 $$1 - 10.2T + 59T^{2}$$
61 $$1 + 13.1T + 61T^{2}$$
67 $$1 + 8.83T + 67T^{2}$$
71 $$1 + 1.14T + 71T^{2}$$
73 $$1 - 0.195T + 73T^{2}$$
79 $$1 + 7.20T + 79T^{2}$$
83 $$1 - 14.9T + 83T^{2}$$
89 $$1 - 1.55T + 89T^{2}$$
97 $$1 - 5.29T + 97T^{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

−10.84883308003732015338994740309, −9.615531540883595798960937986243, −8.634197104267576725249431141829, −7.44731080190493368048680785830, −6.38468311987805840182677202037, −5.99442891828165287416195095325, −4.82369108770828682500824239274, −3.88994231302576247410678738790, −3.22355982813605523705427438329, −1.73194096264220742057972577275, 1.73194096264220742057972577275, 3.22355982813605523705427438329, 3.88994231302576247410678738790, 4.82369108770828682500824239274, 5.99442891828165287416195095325, 6.38468311987805840182677202037, 7.44731080190493368048680785830, 8.634197104267576725249431141829, 9.615531540883595798960937986243, 10.84883308003732015338994740309