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Properties

Label 1-273-273.137-r0-0-0
Degree $1$
Conductor $273$
Sign $0.467 - 0.884i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  − i·2-s − 4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + 16-s + 17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + 16-s + 17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.467 - 0.884i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.467 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099561524 - 0.6626891048i\)
\(L(\frac12)\) \(\approx\) \(1.099561524 - 0.6626891048i\)
\(L(1)\) \(\approx\) \(1.022246720 - 0.4521124117i\)
\(L(1)\) \(\approx\) \(1.022246720 - 0.4521124117i\)

Euler product

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

−25.64866749691396648782616207720, −25.05679343281774198821845511032, −24.219111071993023368049791634396, −23.24334595771276531374362409158, −22.50390605680784127309143696303, −21.27132633909329870014404076598, −20.73018524664554796168165064401, −19.18795269797217259770033989691, −18.22795609692269905156125150537, −17.48359069064794454143361464083, −16.633671752350175262655154549635, −15.794644771995177054761388744826, −14.75111253353715947984983224911, −13.792118022849117444672411756334, −13.05961871792550738895780936987, −12.08481640483068756406321097205, −10.21494452239644998829942042736, −9.68115738216260241004557531657, −8.48397966739745587860469592667, −7.59963949261040226084558260292, −6.402038939061623988753439343809, −5.38476201537949865906873211336, −4.70354882181372960382639818504, −3.03331695775066431359559983520, −1.23908350650803271075532829183, 1.174208409234996941570880625873, 2.60187054619853405935771289876, 3.30319328387998556320469079792, 4.977352398901112874678797533564, 5.756607771624054893979100817349, 7.29998525989811394721760076871, 8.59108912705746421834156965183, 9.64046726784996051489476115105, 10.42957958974115161469078156971, 11.2315500513745167162974660094, 12.403481485755641389634352868585, 13.42256064579587704102590037837, 14.00750324331314284762593759950, 15.0995919625477879212924435952, 16.55329635560066490377329353301, 17.58591589263244596906720933303, 18.403997021509209377414122570519, 19.02047939619335977642487544985, 20.21161576349697919777340982399, 21.18929965539612511325192272117, 21.61496384593950530920607894026, 22.67891121613309070607328077961, 23.435646496707739004008453071398, 24.65626342703536766571106960788, 25.84185731226593285438365877710

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