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Properties

Label 1-3e3-27.7-r0-0-0
Degree $1$
Conductor $27$
Sign $0.727 - 0.686i$
Analytic cond. $0.125387$
Root an. cond. $0.125387$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)13-s + (0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(0.125387\)
Root analytic conductor: \(0.125387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (0:\ ),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8490271807 - 0.3372969165i\)
\(L(\frac12)\) \(\approx\) \(0.8490271807 - 0.3372969165i\)
\(L(1)\) \(\approx\) \(1.115760427 - 0.3412711769i\)
\(L(1)\) \(\approx\) \(1.115760427 - 0.3412711769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−38.37006695871833379169365588964, −36.274620460130378984339463788882, −35.402034786379264472974615654630, −34.02887553914528176727145001360, −32.98450940829158470999413590497, −31.66519302300073343520532810926, −30.78390994133351781706870369410, −29.43613779766801443137720978188, −27.499378897128436615294809322413, −26.36976147454807292943786649772, −24.93949048412501969405726267577, −23.414775503556971784434443983558, −23.095527271012988967278063217144, −21.0423512759974302105356328023, −20.03598160893936930548290354701, −17.893140159843779489600389111051, −16.34456156505821516520485918029, −15.40254334374809065819309872172, −13.76428114525284607594016530240, −12.510859478141777837115129966788, −10.90287068662359416861863466120, −8.27354959631020868239674358954, −7.15829115002224906226372528073, −5.03532086681002160373466118417, −3.58950910323227086925668585890, 2.65531742487320095547138102958, 4.44438357855947998471443120743, 6.29164786968591117600241502764, 8.558903351798499135324663408425, 10.72996026014505713935126654307, 11.787834728111295578053700052211, 13.18995949240848713693921055338, 14.93299902446899029176490644766, 15.81171109138441681210955252200, 18.45614389719719568044762940434, 19.322195944851260909865068085509, 20.913104195132228126122931810409, 22.03107949704272862435790170247, 23.38976133549368094409861490292, 24.32369897415381571503674604675, 26.249945738646348600298867042345, 27.91319882342413445836187070183, 28.74522149472364032448013532545, 30.51271429056067775742478794253, 31.16979786221864763223296122943, 32.27545631674212666868772665971, 33.89173526726370811178011254070, 34.96354881398418543688449087082, 36.761480619543136652806529917104, 38.05705172963086020231747861133

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