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Properties

Label 2-3e6-27.16-c1-0-20
Degree $2$
Conductor $729$
Sign $-0.0581 + 0.998i$
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 + (−0.173 + 0.984i)7-s + (3.83 − 3.21i)13-s + (−3.75 + 1.36i)16-s + (3.5 − 6.06i)19-s + (−3.83 − 3.21i)25-s + 1.99·28-s + (−0.694 − 3.93i)31-s + (−5.5 − 9.52i)37-s + (−7.51 + 2.73i)43-s + (5.63 + 2.05i)49-s + (−7.66 − 6.42i)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 + (−0.0656 + 0.372i)7-s + (1.06 − 0.891i)13-s + (−0.939 + 0.342i)16-s + (0.802 − 1.39i)19-s + (−0.766 − 0.642i)25-s + 0.377·28-s + (−0.124 − 0.707i)31-s + (−0.904 − 1.56i)37-s + (−1.14 + 0.417i)43-s + (0.805 + 0.293i)49-s + (−1.06 − 0.891i)52-s + (−0.0222 + 0.126i)61-s + (0.5 + 0.866i)64-s + (0.467 − 0.392i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.0581 + 0.998i$
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: \((2,\ 729,\ (\ :1/2),\ -0.0581 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899832 - 0.953766i\)
\(L(\frac12)\) \(\approx\) \(0.899832 - 0.953766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-3.83 + 3.21i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.694 + 3.93i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.51 - 2.73i)T + (32.9 - 27.6i)T^{2} \)
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 \( 1 + (-13.0 - 10.9i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-17.8 + 6.49i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.18587568191887176973264987334, −9.321743862294825133484861837392, −8.658286509471773915480666684529, −7.56559368398209851824218252662, −6.43692348135973875668567307408, −5.69174325100426903541388450020, −4.90893828520675537275892166809, −3.61496372560854211377731395538, −2.23758673685542541862360751672, −0.71330027533948425462725648783, 1.65923592381617382130211192189, 3.36275353103461006961630319987, 3.89218635874035959723193267001, 5.11962502753319739703464465246, 6.35789006419460820558980929749, 7.20620617462523893619729132217, 8.088149586737844897680918955967, 8.773172659361108804966851889868, 9.697064431151749629723910901010, 10.61513327210874298667115020417

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