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Properties

Label 2-3e3-3.2-c4-0-3
Degree $2$
Conductor $27$
Sign $i$
Analytic cond. $2.79098$
Root an. cond. $1.67062$
Motivic weight $4$
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 + 7·4-s − 33i·5-s − 19·7-s − 69i·8-s − 99·10-s + 123i·11-s + 302·13-s + 57i·14-s − 95·16-s + 414i·17-s − 304·19-s − 231i·20-s + 369·22-s + 300i·23-s + ⋯
L(s)  = 1  − 0.750i·2-s + 0.437·4-s − 1.32i·5-s − 0.387·7-s − 1.07i·8-s − 0.989·10-s + 1.01i·11-s + 1.78·13-s + 0.290i·14-s − 0.371·16-s + 1.43i·17-s − 0.842·19-s − 0.577i·20-s + 0.762·22-s + 0.567i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $i$
Analytic conductor: \(2.79098\)
Root analytic conductor: \(1.67062\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :2),\ i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.05829 - 1.05829i\)
\(L(\frac12)\) \(\approx\) \(1.05829 - 1.05829i\)
\(L(3)\) 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 \( 1 + 3iT - 16T^{2} \)
5 \( 1 + 33iT - 625T^{2} \)
7 \( 1 + 19T + 2.40e3T^{2} \)
11 \( 1 - 123iT - 1.46e4T^{2} \)
13 \( 1 - 302T + 2.85e4T^{2} \)
17 \( 1 - 414iT - 8.35e4T^{2} \)
19 \( 1 + 304T + 1.30e5T^{2} \)
23 \( 1 - 300iT - 2.79e5T^{2} \)
29 \( 1 + 678iT - 7.07e5T^{2} \)
31 \( 1 - 239T + 9.23e5T^{2} \)
37 \( 1 - 740T + 1.87e6T^{2} \)
41 \( 1 - 228iT - 2.82e6T^{2} \)
43 \( 1 + 982T + 3.41e6T^{2} \)
47 \( 1 - 2.16e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.59e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.92e3iT - 1.21e7T^{2} \)
61 \( 1 + 316T + 1.38e7T^{2} \)
67 \( 1 - 4.62e3T + 2.01e7T^{2} \)
71 \( 1 - 1.81e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.03e3T + 2.83e7T^{2} \)
79 \( 1 + 1.04e4T + 3.89e7T^{2} \)
83 \( 1 - 1.26e4iT - 4.74e7T^{2} \)
89 \( 1 + 7.00e3iT - 6.27e7T^{2} \)
97 \( 1 + 6.51e3T + 8.85e7T^{2} \)
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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

−16.23279407038076774837417964119, −15.31650150181296386185682261607, −13.11924741795611269714341046009, −12.58931403462131941534971713157, −11.18386426520262237837112012986, −9.810941832294436441718305423684, −8.356504099295465831842273148100, −6.25477143298326408900515905722, −4.03088518790966787447861190872, −1.50186504539961395047944735848, 3.05553397015761697482722952315, 6.04368205348361137920981572860, 6.92663286302957008782946200185, 8.537052936300178784252267315297, 10.66317134386680375165190413963, 11.42635878253882303189727707945, 13.57193457608714596607695177070, 14.57240935625533123466617305303, 15.76107984422775803842044476115, 16.50259104642843069262332811953

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