# Properties

 Label 2-3e6-27.22-c1-0-26 Degree $2$ Conductor $729$ Sign $-0.286 + 0.957i$ 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 + (−0.347 + 1.96i)4-s + (−0.694 − 3.93i)7-s + (−5.36 − 4.49i)13-s + (−3.75 − 1.36i)16-s + (0.5 + 0.866i)19-s + (−3.83 + 3.21i)25-s + 7.99·28-s + (1.91 − 10.8i)31-s + (5 − 8.66i)37-s + (−4.69 − 1.71i)43-s + (−8.45 + 3.07i)49-s + (10.7 − 8.99i)52-s + (−0.173 − 0.984i)61-s + (4 − 6.92i)64-s + (3.83 + 3.21i)67-s + ⋯
 L(s)  = 1 + (−0.173 + 0.984i)4-s + (−0.262 − 1.48i)7-s + (−1.48 − 1.24i)13-s + (−0.939 − 0.342i)16-s + (0.114 + 0.198i)19-s + (−0.766 + 0.642i)25-s + 1.51·28-s + (0.343 − 1.94i)31-s + (0.821 − 1.42i)37-s + (−0.716 − 0.260i)43-s + (−1.20 + 0.439i)49-s + (1.48 − 1.24i)52-s + (−0.0222 − 0.126i)61-s + (0.5 − 0.866i)64-s + (0.467 + 0.392i)67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$729$$    =    $$3^{6}$$ Sign: $-0.286 + 0.957i$ Analytic conductor: $$5.82109$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{729} (325, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 729,\ (\ :1/2),\ -0.286 + 0.957i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.446000 - 0.599083i$$ $$L(\frac12)$$ $$\approx$$ $$0.446000 - 0.599083i$$ $$L(\frac{3}{2})$$ 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 + (0.347 - 1.96i)T^{2}$$
5 $$1 + (3.83 - 3.21i)T^{2}$$
7 $$1 + (0.694 + 3.93i)T + (-6.57 + 2.39i)T^{2}$$
11 $$1 + (8.42 + 7.07i)T^{2}$$
13 $$1 + (5.36 + 4.49i)T + (2.25 + 12.8i)T^{2}$$
17 $$1 + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-21.6 - 7.86i)T^{2}$$
29 $$1 + (5.03 - 28.5i)T^{2}$$
31 $$1 + (-1.91 + 10.8i)T + (-29.1 - 10.6i)T^{2}$$
37 $$1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (7.11 + 40.3i)T^{2}$$
43 $$1 + (4.69 + 1.71i)T + (32.9 + 27.6i)T^{2}$$
47 $$1 + (-44.1 + 16.0i)T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + (45.1 - 37.9i)T^{2}$$
61 $$1 + (0.173 + 0.984i)T + (-57.3 + 20.8i)T^{2}$$
67 $$1 + (-3.83 - 3.21i)T + (11.6 + 65.9i)T^{2}$$
71 $$1 + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (9.95 - 8.35i)T + (13.7 - 77.7i)T^{2}$$
83 $$1 + (14.4 - 81.7i)T^{2}$$
89 $$1 + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (4.69 + 1.71i)T + (74.3 + 62.3i)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

−9.996477807193794753705179535360, −9.504665808894277382728136680681, −8.074170725257147030203583753843, −7.60037640200099998854790644840, −6.98567091502667497329450257378, −5.58629501134884344251630837262, −4.39226810190590639148762898278, −3.65892967745008029953096080381, −2.53487223280189715835326118140, −0.36754890654905086576193349311, 1.81409665349745421782827441751, 2.78895331319813385966778326699, 4.57828893963226823238894653893, 5.20201244562254585610395849962, 6.22869250899549937937463422121, 6.88696430194577783622817440697, 8.275484792753622714708764787517, 9.180583112405194193077904577244, 9.646660494873775946032257249295, 10.45429476993624832380300033315