# Properties

 Label 2-3e6-9.4-c1-0-21 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 no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.984 − 1.70i)2-s + (−0.939 + 1.62i)4-s + (1.85 − 3.20i)5-s + (1.17 + 2.03i)7-s − 0.237·8-s − 7.29·10-s + (−1.08 − 1.88i)11-s + (2.35 − 4.08i)13-s + (2.31 − 4.00i)14-s + (2.11 + 3.66i)16-s + 2.93·17-s − 6.22·19-s + (3.47 + 6.02i)20-s + (−2.14 + 3.71i)22-s + (0.259 − 0.449i)23-s + ⋯
 L(s)  = 1 + (−0.696 − 1.20i)2-s + (−0.469 + 0.813i)4-s + (0.827 − 1.43i)5-s + (0.443 + 0.768i)7-s − 0.0839·8-s − 2.30·10-s + (−0.328 − 0.568i)11-s + (0.654 − 1.13i)13-s + (0.617 − 1.07i)14-s + (0.528 + 0.915i)16-s + 0.711·17-s − 1.42·19-s + (0.777 + 1.34i)20-s + (−0.457 + 0.792i)22-s + (0.0541 − 0.0937i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\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: $\chi_{729} (244, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.09970i$$ $$L(\frac12)$$ $$\approx$$ $$1.09970i$$ $$L(\frac{3}{2})$$ 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 + (0.984 + 1.70i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-1.85 + 3.20i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + (-1.17 - 2.03i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (1.08 + 1.88i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (-2.35 + 4.08i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 - 2.93T + 17T^{2}$$
19 $$1 + 6.22T + 19T^{2}$$
23 $$1 + (-0.259 + 0.449i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (-2.15 + 3.72i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 2.41T + 37T^{2}$$
41 $$1 + (-1.24 + 2.16i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.532 + 0.921i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (0.118 + 0.205i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 4.66T + 53T^{2}$$
59 $$1 + (6.65 - 11.5i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-1.83 - 3.18i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (7.14 - 12.3i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 1.20T + 71T^{2}$$
73 $$1 + 4.68T + 73T^{2}$$
79 $$1 + (-6.40 - 11.0i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (5.65 + 9.78i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + 0.699T + 89T^{2}$$
97 $$1 + (-3.54 - 6.13i)T + (-48.5 + 84.0i)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

−10.02946403472072155405919370059, −9.121994724563930969890079568869, −8.539292526321715990636444872386, −8.061097215311714611062430662636, −5.88799114522623059189722411254, −5.66270878049593023214333996607, −4.30042438692821367574023599096, −2.84274357397700214846368122985, −1.82545267961753784815488932767, −0.72657846915495905354737958519, 1.83876358888238986358815443859, 3.27315380605514617259578191652, 4.68783395964814139339494249893, 6.05395578117833403698671848018, 6.56033652583126607303429966780, 7.25297311019405750819793495838, 7.991951455583657035487670697358, 9.056670494994224602220298480244, 9.860869380236037278176401275307, 10.60683512157953762822604188156