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Properties

Label 1-273-273.188-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.999 + 0.0386i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·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 + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 25-s + (0.5 + 0.866i)29-s + i·31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·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 + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 25-s + (0.5 + 0.866i)29-s + i·31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0386i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.999 + 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04035937202 - 2.090154684i\)
\(L(\frac12)\) \(\approx\) \(-0.04035937202 - 2.090154684i\)
\(L(1)\) \(\approx\) \(1.142511047 - 0.9445516671i\)
\(L(1)\) \(\approx\) \(1.142511047 - 0.9445516671i\)

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 \( 1 - iT \)
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 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 - T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
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

−25.8903189729649702934171832710, −25.08841745733967128570607184441, −23.78856952038852424594170716780, −23.39460781179878197035269359924, −22.35262568355725411756340833778, −21.55052594809824347500882117978, −20.90372564778847595170859937394, −19.51773396802065500439863846581, −18.59168748137642923493290292344, −17.52161667050207724113556083823, −16.57042210443212332302173523904, −15.45126835907709552914973062697, −14.881897580502660560966869950959, −13.857181506899931615903293673728, −13.0974961440521542742687976608, −11.92235332618066869546910925543, −10.97734640777054819955601207765, −10.03880643590580826008112156166, −8.24429202136998856386645770933, −7.59973988982645103252811489795, −6.30781994759845765386630441747, −5.710035421223637165254089135609, −4.210253822347712105638531200387, −3.22465978026250118491136957600, −2.15547301978563237417902420441, 0.43864108055828928597567190432, 1.83823391528615786593574860935, 3.0326297584357460780678097199, 4.52953252864850341941574356277, 5.039705938338958094110578623097, 6.2642801087330766222417268003, 7.566406369080737538440386694062, 8.88325195261076337524303318671, 9.97936929871990584890371483894, 10.918600752591499659722907598192, 12.19200042071693089345403596411, 12.684113091176534152696830954996, 13.63141952899847365427045737594, 14.622954011341790974488415108142, 15.76012917452043688250424686022, 16.36854299845150968589378551168, 17.737474239977562607813634505232, 18.80990579364443166330236154925, 19.91794246558245409001081937785, 20.55969492508469671516644711042, 21.27820967255589360805463267789, 22.223615791973224833852115764, 23.46768129756314232810163532144, 23.72220301028786219276669171570, 24.9305941250679068861687956184

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