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Properties

Label 1-273-273.212-r1-0-0
Degree $1$
Conductor $273$
Sign $0.564 - 0.825i$
Analytic cond. $29.3379$
Root an. cond. $29.3379$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + 10-s + 11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + 10-s + 11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8819341296 - 0.4653268246i\)
\(L(\frac12)\) \(\approx\) \(0.8819341296 - 0.4653268246i\)
\(L(1)\) \(\approx\) \(0.7583444334 + 0.05789709291i\)
\(L(1)\) \(\approx\) \(0.7583444334 + 0.05789709291i\)

Euler product

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

−25.71181116151348054706350363262, −25.095432606662503456674514675125, −23.44677172651425366260238153590, −22.772290644561357382427980618033, −21.88949554206472251228720621681, −21.082138613365559057092666532777, −19.88872386470319889329660957381, −19.27617730501312695606275090922, −18.51163665033830297817737205461, −17.52340421548943314060726234911, −16.658593570839385830119814245228, −15.41593017072431168328647834737, −14.33244243486305308450421511262, −13.40268547443474080897909876010, −12.02282765583535787371922653104, −11.550227330523582842100370796822, −10.49895609279726605005053907028, −9.60723516354167951372912890420, −8.50212644519929764074540633864, −7.45416079503406440535319701225, −6.46544033661574964305095491218, −4.62505554896584718873964462151, −3.55954162183579400198905872921, −2.623904011600699088046897437286, −1.139861515204823565484986868834, 0.43869163195780131983004943586, 1.64363853341555319144378078004, 3.884873734850229953257999746494, 4.79063206112297378336486174737, 6.00876681130680623681587602724, 6.98404363094788005753607767331, 8.25120077863334172641934014499, 8.76656849049971358180349681774, 9.860313769468694489044067248931, 11.02518779358766508329487826802, 12.29480501647473838006375906513, 13.22130935365804884365279066544, 14.535715379330034662071561118259, 15.13429573793281706712826057723, 16.38940628919382561060238858832, 16.825941379708416322016543360781, 17.72862534376169060280763800362, 19.06344275453956226931043831958, 19.54867005083728987837475202800, 20.59811682916236659750014874937, 21.83318481023663330791002718791, 23.00703599402963907141251559958, 23.67634708521820315641511505676, 24.579894690868998063130995490752, 25.196720640410879992540979317816

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