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Properties

Label 2-3e6-27.4-c1-0-0
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  + (−2.37 − 0.866i)2-s + (3.37 + 2.83i)4-s + (0.0812 − 0.460i)5-s + (−2.47 + 2.07i)7-s + (−3.05 − 5.28i)8-s + (−0.592 + 1.02i)10-s + (−0.539 − 3.05i)11-s + (2.05 − 0.747i)13-s + (7.67 − 2.79i)14-s + (1.15 + 6.53i)16-s + (−1.5 + 2.59i)17-s + (−0.0209 − 0.0362i)19-s + (1.58 − 1.32i)20-s + (−1.36 + 7.74i)22-s + (4.67 + 3.92i)23-s + ⋯
L(s)  = 1  + (−1.68 − 0.612i)2-s + (1.68 + 1.41i)4-s + (0.0363 − 0.206i)5-s + (−0.934 + 0.783i)7-s + (−1.07 − 1.86i)8-s + (−0.187 + 0.324i)10-s + (−0.162 − 0.922i)11-s + (0.569 − 0.207i)13-s + (2.05 − 0.746i)14-s + (0.288 + 1.63i)16-s + (−0.363 + 0.630i)17-s + (−0.00480 − 0.00832i)19-s + (0.353 − 0.296i)20-s + (−0.291 + 1.65i)22-s + (0.975 + 0.818i)23-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: $\chi_{729} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ -0.0581 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185860 + 0.197000i\)
\(L(\frac12)\) \(\approx\) \(0.185860 + 0.197000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.37 + 0.866i)T + (1.53 + 1.28i)T^{2} \)
5 \( 1 + (-0.0812 + 0.460i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (2.47 - 2.07i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.539 + 3.05i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-2.05 + 0.747i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0209 + 0.0362i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.67 - 3.92i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (6.17 + 2.24i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.76 + 4.00i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (1.79 - 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.23 - 2.63i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.102 + 0.579i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.40 - 6.20i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 4.95T + 53T^{2} \)
59 \( 1 + (1.48 - 8.40i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (0.971 - 0.815i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (9.40 - 3.42i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.11 - 7.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 + 3.77i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.41 - 0.516i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-7.93 - 13.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.23 - 18.3i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.53062875453819822716505715867, −9.578726260124383193940488252280, −8.968629371050873365898862125325, −8.463513422206343942328544660947, −7.45357357108776627212413341619, −6.44284519321611044766302346366, −5.53396240141280029888023670253, −3.54457905232405511952809491081, −2.75020175818702484396726527528, −1.35313326382164775267940421347, 0.25249697471821080337441353075, 1.82140077763391736019569406286, 3.33975228418638517137297466404, 4.93282899135714454151038459976, 6.31996426772637409444374699483, 7.02195835893292388250335393097, 7.35708462378287347570092244614, 8.687844077203152640663428598585, 9.171354038613052615186793528558, 10.10810178173060739706104918166

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