# Properties

 Label 2-3e6-81.67-c1-0-30 Degree $2$ Conductor $729$ Sign $-0.659 + 0.751i$ 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 + (1.97 − 1.29i)2-s + (1.41 − 3.28i)4-s + (1.82 − 1.93i)5-s + (−4.56 − 0.533i)7-s + (−0.647 − 3.67i)8-s + (1.09 − 6.18i)10-s + (−0.138 + 0.463i)11-s + (0.253 − 4.34i)13-s + (−9.69 + 4.87i)14-s + (−1.13 − 1.20i)16-s + (0.936 − 0.340i)17-s + (0.818 + 0.297i)19-s + (−3.77 − 8.74i)20-s + (0.327 + 1.09i)22-s + (2.03 − 0.238i)23-s + ⋯
 L(s)  = 1 + (1.39 − 0.917i)2-s + (0.708 − 1.64i)4-s + (0.816 − 0.865i)5-s + (−1.72 − 0.201i)7-s + (−0.229 − 1.29i)8-s + (0.344 − 1.95i)10-s + (−0.0418 + 0.139i)11-s + (0.0702 − 1.20i)13-s + (−2.59 + 1.30i)14-s + (−0.283 − 0.300i)16-s + (0.227 − 0.0826i)17-s + (0.187 + 0.0683i)19-s + (−0.843 − 1.95i)20-s + (0.0699 + 0.233i)22-s + (0.425 − 0.0496i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $-0.659 + 0.751i$ 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: $$(2,\ 729,\ (\ :1/2),\ -0.659 + 0.751i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.26816 - 2.79999i$$ $$L(\frac12)$$ $$\approx$$ $$1.26816 - 2.79999i$$ $$L(\frac{3}{2})$$ 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 + (-1.97 + 1.29i)T + (0.792 - 1.83i)T^{2}$$
5 $$1 + (-1.82 + 1.93i)T + (-0.290 - 4.99i)T^{2}$$
7 $$1 + (4.56 + 0.533i)T + (6.81 + 1.61i)T^{2}$$
11 $$1 + (0.138 - 0.463i)T + (-9.19 - 6.04i)T^{2}$$
13 $$1 + (-0.253 + 4.34i)T + (-12.9 - 1.50i)T^{2}$$
17 $$1 + (-0.936 + 0.340i)T + (13.0 - 10.9i)T^{2}$$
19 $$1 + (-0.818 - 0.297i)T + (14.5 + 12.2i)T^{2}$$
23 $$1 + (-2.03 + 0.238i)T + (22.3 - 5.30i)T^{2}$$
29 $$1 + (-0.741 - 0.372i)T + (17.3 + 23.2i)T^{2}$$
31 $$1 + (-2.39 - 3.21i)T + (-8.89 + 29.6i)T^{2}$$
37 $$1 + (-0.840 - 0.704i)T + (6.42 + 36.4i)T^{2}$$
41 $$1 + (0.244 + 0.160i)T + (16.2 + 37.6i)T^{2}$$
43 $$1 + (-10.9 + 2.58i)T + (38.4 - 19.2i)T^{2}$$
47 $$1 + (2.41 - 3.23i)T + (-13.4 - 45.0i)T^{2}$$
53 $$1 + (-0.806 + 1.39i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (1.51 + 5.04i)T + (-49.2 + 32.4i)T^{2}$$
61 $$1 + (-1.59 - 3.70i)T + (-41.8 + 44.3i)T^{2}$$
67 $$1 + (12.8 - 6.47i)T + (40.0 - 53.7i)T^{2}$$
71 $$1 + (2.20 - 12.4i)T + (-66.7 - 24.2i)T^{2}$$
73 $$1 + (-1.00 - 5.71i)T + (-68.5 + 24.9i)T^{2}$$
79 $$1 + (-5.40 + 3.55i)T + (31.2 - 72.5i)T^{2}$$
83 $$1 + (10.6 - 6.97i)T + (32.8 - 76.2i)T^{2}$$
89 $$1 + (2.74 + 15.5i)T + (-83.6 + 30.4i)T^{2}$$
97 $$1 + (1.39 + 1.47i)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

−10.11579583649572479640825062837, −9.688848220137294913718499044586, −8.627644043616749204638192360411, −7.11613771432058975279116286035, −5.96005806315539748051348692018, −5.56714822853526977098969726376, −4.49023149725094763958122498064, −3.35953432300085113604651727442, −2.65825808768282309425370533553, −1.06630691782687747901209926944, 2.54686336904745937293220918529, 3.32351967778801385814103434496, 4.33602113990173296530092496293, 5.65552927926699501501420129628, 6.32603688421399358990403741573, 6.68631617182414043475814220412, 7.58050314455129282003399774714, 9.122520345132817829549190751426, 9.752918749762258520610903406018, 10.72238286435491220050229819752