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Properties

Label 2-3e6-27.16-c1-0-16
Degree $2$
Conductor $729$
Sign $-0.597 - 0.802i$
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  + (2.07 + 1.73i)2-s + (0.923 + 5.23i)4-s + (1.57 + 0.571i)5-s + (0.0869 − 0.492i)7-s + (−4.48 + 7.77i)8-s + (2.26 + 3.91i)10-s + (1.80 − 0.655i)11-s + (2.38 − 2.00i)13-s + (1.03 − 0.870i)14-s + (−12.8 + 4.66i)16-s + (−1.33 − 2.30i)17-s + (−2.89 + 5.02i)19-s + (−1.54 + 8.75i)20-s + (4.87 + 1.77i)22-s + (−0.806 − 4.57i)23-s + ⋯
L(s)  = 1  + (1.46 + 1.22i)2-s + (0.461 + 2.61i)4-s + (0.702 + 0.255i)5-s + (0.0328 − 0.186i)7-s + (−1.58 + 2.74i)8-s + (0.715 + 1.23i)10-s + (0.543 − 0.197i)11-s + (0.661 − 0.554i)13-s + (0.277 − 0.232i)14-s + (−3.20 + 1.16i)16-s + (−0.323 − 0.559i)17-s + (−0.664 + 1.15i)19-s + (−0.345 + 1.95i)20-s + (1.03 + 0.378i)22-s + (−0.168 − 0.954i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $-0.597 - 0.802i$
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: \((2,\ 729,\ (\ :1/2),\ -0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65146 + 3.28834i\)
\(L(\frac12)\) \(\approx\) \(1.65146 + 3.28834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 1.73i)T + (0.347 + 1.96i)T^{2} \)
5 \( 1 + (-1.57 - 0.571i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.0869 + 0.492i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.80 + 0.655i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.38 + 2.00i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.33 + 2.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.89 - 5.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.806 + 4.57i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.00 + 1.68i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.801 + 4.54i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-2.42 - 4.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.84 - 7.42i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-8.46 + 3.07i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.18 + 6.72i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 5.43T + 53T^{2} \)
59 \( 1 + (-2.05 - 0.749i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.18 + 6.73i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.56 + 8.02i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.41 - 2.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.96 - 8.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.06 - 3.41i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.08 + 1.75i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (5.60 - 9.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.47 + 2.35i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.91288220768177222327093682243, −9.777111889340637733828984794453, −8.521096665369831330295473457473, −7.920443041128653527850722953193, −6.77303699825183306031888967445, −6.19488415953359441722550050847, −5.54470596293491390194168222664, −4.38478859748851329555766620471, −3.59746338482290027350187957423, −2.35406609192328040647337958184, 1.44374389400836513838790960303, 2.28075391677472717013559684606, 3.59962065232048837628170889475, 4.39476783795046470490191158822, 5.41376707308096796742638046360, 6.10505616040013816913471212464, 7.00017166368174031798947974233, 8.920767225902155506187733913145, 9.404386467093150675872362064686, 10.45941374787152416487592270000

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