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Properties

Label 1-273-273.86-r0-0-0
Degree $1$
Conductor $273$
Sign $-0.228 - 0.973i$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − 29-s + (−0.866 − 0.5i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − 29-s + (−0.866 − 0.5i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ -0.228 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1384481016 - 0.1747215720i\)
\(L(\frac12)\) \(\approx\) \(0.1384481016 - 0.1747215720i\)
\(L(1)\) \(\approx\) \(0.4728044996 + 0.05165524884i\)
\(L(1)\) \(\approx\) \(0.4728044996 + 0.05165524884i\)

Euler product

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

−26.23264054807429082446418335026, −25.17623179219845827167236597566, −24.305910684992382906672463590257, −23.26286743629270170287817622986, −22.25974963008623495594219845212, −21.06447022396882074623660610558, −20.288876596827213066188697348867, −19.740206148056630106214704379163, −18.54936422507537405996765784437, −17.97693525992391407116878463335, −16.732926420499149548271580337928, −15.96879754029416812837686914591, −15.25368695056640348542630706312, −13.56687918393541407188968315126, −12.5446516988810164903898402576, −11.76712012743147427469677835687, −10.886979616317685126906289065701, −9.73906182595495925290531840492, −8.86250671105621346877254974564, −7.71834558828274402690843507452, −7.22459841953591666088931707630, −5.3715946660724320805527219427, −4.0620907122572011897919521817, −2.93973059998581400213878519333, −1.50105277899594920176994817254, 0.19935103304320804136659622195, 2.12672032903600838799574367339, 3.50078477970898169585898134219, 5.05666933192917359743388715937, 6.25135354160131653508309085737, 7.30662066621483252611521726692, 8.08276168030489992000111829811, 9.00515938523282610990681737563, 10.38226082462544931404182908765, 10.96759713729480678707092589244, 12.004674719052664754740801296154, 13.45195906752044339108162135230, 14.64404268371148944668690666793, 15.44814126506171684977630997710, 16.129545963684627785780756397852, 17.13670972289072969645820341691, 18.34659666932727930012404292502, 18.74968233272313176715965781009, 19.82781366000742838366532924209, 20.51378285133094328060255232692, 21.94935910294320990377626844103, 22.93383509287088025661551231283, 24.05825720526071262302637128774, 24.263316857758589621739802313281, 25.85861832660038591410651999532

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