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Properties

Label 2-3e3-3.2-c2-0-2
Degree $2$
Conductor $27$
Sign $i$
Analytic cond. $0.735696$
Root an. cond. $0.857727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·2-s − 5·4-s + 3i·5-s + 5·7-s + 3i·8-s + 9·10-s + 15i·11-s − 10·13-s − 15i·14-s − 11·16-s − 18i·17-s − 16·19-s − 15i·20-s + 45·22-s + 12i·23-s + ⋯
L(s)  = 1  − 1.5i·2-s − 1.25·4-s + 0.600i·5-s + 0.714·7-s + 0.375i·8-s + 0.900·10-s + 1.36i·11-s − 0.769·13-s − 1.07i·14-s − 0.687·16-s − 1.05i·17-s − 0.842·19-s − 0.750i·20-s + 2.04·22-s + 0.521i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $i$
Analytic conductor: \(0.735696\)
Root analytic conductor: \(0.857727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.670334 - 0.670334i\)
\(L(\frac12)\) \(\approx\) \(0.670334 - 0.670334i\)
\(L(2)\) 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 \( 1 + 3iT - 4T^{2} \)
5 \( 1 - 3iT - 25T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 - 15iT - 121T^{2} \)
13 \( 1 + 10T + 169T^{2} \)
17 \( 1 + 18iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 - 12iT - 529T^{2} \)
29 \( 1 + 30iT - 841T^{2} \)
31 \( 1 + T + 961T^{2} \)
37 \( 1 - 20T + 1.36e3T^{2} \)
41 \( 1 + 60iT - 1.68e3T^{2} \)
43 \( 1 - 50T + 1.84e3T^{2} \)
47 \( 1 - 6iT - 2.20e3T^{2} \)
53 \( 1 - 27iT - 2.80e3T^{2} \)
59 \( 1 - 30iT - 3.48e3T^{2} \)
61 \( 1 + 76T + 3.72e3T^{2} \)
67 \( 1 + 10T + 4.48e3T^{2} \)
71 \( 1 - 90iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 - 14T + 6.24e3T^{2} \)
83 \( 1 + 3iT - 6.88e3T^{2} \)
89 \( 1 + 90iT - 7.92e3T^{2} \)
97 \( 1 + 85T + 9.40e3T^{2} \)
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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

−17.36808523137500384684965942861, −15.32702670319768498329699627984, −14.14876383614042419244364506857, −12.65414210033438109237929814485, −11.65390726870685934002347607507, −10.52950361117322614987594461701, −9.420708282430980154915816688071, −7.28542109883854820063536672271, −4.53354833528689371647994175977, −2.36734171042264431836895026524, 4.83051333614660063586439682390, 6.24922237424642260720064150063, 7.981613995047342787015509059394, 8.829029835700150106568585551216, 10.99282694792126830280638291561, 12.78777173527289554999790494628, 14.21316958405189901201418238593, 15.02205089555784189285554512850, 16.44271949144973078182113659631, 16.95867641883435699347127869064

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