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Properties

Label 2-3e6-27.13-c1-0-18
Degree $2$
Conductor $729$
Sign $0.893 + 0.448i$
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  + (1.87 − 0.684i)4-s + (0.939 + 0.342i)7-s + (0.868 − 4.92i)13-s + (3.06 − 2.57i)16-s + (3.5 + 6.06i)19-s + (−0.868 − 4.92i)25-s + 2·28-s + (3.75 − 1.36i)31-s + (−5.5 + 9.52i)37-s + (6.12 − 5.14i)43-s + (−4.59 − 3.85i)49-s + (−1.73 − 9.84i)52-s + (0.939 + 0.342i)61-s + (4.00 − 6.92i)64-s + (0.868 − 4.92i)67-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)4-s + (0.355 + 0.129i)7-s + (0.240 − 1.36i)13-s + (0.766 − 0.642i)16-s + (0.802 + 1.39i)19-s + (−0.173 − 0.984i)25-s + 0.377·28-s + (0.675 − 0.245i)31-s + (−0.904 + 1.56i)37-s + (0.934 − 0.784i)43-s + (−0.656 − 0.550i)49-s + (−0.240 − 1.36i)52-s + (0.120 + 0.0437i)61-s + (0.500 − 0.866i)64-s + (0.106 − 0.601i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95479 - 0.463295i\)
\(L(\frac12)\) \(\approx\) \(1.95479 - 0.463295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.868 + 4.92i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-3.75 + 1.36i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-6.12 + 5.14i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.868 + 4.92i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.95 - 16.7i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.5 - 12.2i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.29771174607494653914169656129, −9.818383213622276193531768044828, −8.296594641017158319484057693855, −7.87305415028100094866877835358, −6.75858138093147949467261678963, −5.85837831947177063687722787420, −5.15651556945074164642761987560, −3.62090610091713417808477378267, −2.55910998280373179008796778097, −1.21879467525710883604211949684, 1.53827353629654300169931148094, 2.72184952383379176943349457570, 3.89140524968830114346084977688, 5.02359505910417287553351515004, 6.20666735076104675126060596197, 7.06161528353037875473470237816, 7.62991530431921058117347079900, 8.804880235518497228374776897340, 9.495088441770129377999257989015, 10.74184171131108221650325611893

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