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Properties

Label 2-3e6-27.22-c1-0-30
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: $\chi_{729} (325, \cdot )$
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

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

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