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Properties

Label 2-3e6-27.16-c1-0-2
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 $0$

Origins

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Normalization:  

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: $\chi_{729} (406, \cdot )$
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

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

Graph of the $Z$-function along the critical line