Properties

Label 1-273-273.230-r0-0-0
Degree $1$
Conductor $273$
Sign $-0.252 + 0.967i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056391509 + 1.367615537i\)
\(L(\frac12)\) \(\approx\) \(1.056391509 + 1.367615537i\)
\(L(1)\) \(\approx\) \(1.202132386 + 0.8278040022i\)
\(L(1)\) \(\approx\) \(1.202132386 + 0.8278040022i\)

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 \( 1 + T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
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

−25.12466516619321023307065977013, −24.59757427417644218759631206246, −23.48145581500509637361077287348, −22.32146999213406539951792999559, −21.954509525617879442343786901314, −20.82211243096057944324051000121, −20.31695949753608962117404434787, −18.987774271164617918811715766164, −18.388344040572859649452715647962, −17.31024241156533136063893749064, −16.235782753712106003912853030018, −14.848476875758838922301974855691, −13.95385102338758835708172524732, −13.41808379690220550096183167327, −12.27990265758525294078408923671, −11.31862992836991316951301526563, −10.3428152815003180832846207498, −9.455337974816870880906600593022, −8.56678527179679518212046159687, −6.68275947806258230004709991475, −5.76847922261267732367280370332, −4.7909716816684875382086717122, −3.4301654841135520551571397405, −2.370925522613103881174261715333, −1.12100415854391069225694789742, 1.824816825704960166489948686006, 3.28762061245949480477879410764, 4.6236983765543234116531996381, 5.5287450340521253839243092658, 6.59670560024070417291167078274, 7.363904697776476789573771492631, 8.8518794643006606570948916236, 9.47263800853339197362253895657, 10.84624979770336633813797010875, 12.2677978891128792606202731742, 13.07131272170200320252722398921, 13.937927181308284960607132695238, 14.80458229769957771780827630225, 15.66126518261074046262737914727, 16.8579426276921012504008258877, 17.54994247149948111917823399732, 18.157402809036367270507551275687, 19.67388382202229462616689960684, 20.76687793028697030425460877902, 21.79379271853094569956748433532, 22.22674785654762854607795561296, 23.37349652108373616912254114739, 24.21875610059432672485698346565, 25.16331036623016307331560064627, 25.706813152168143760406640546918

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