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Properties

Label 1-273-273.95-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.455 + 0.890i$
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  + 2-s + 4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 16-s − 17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 16-s − 17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.455 + 0.890i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.455 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333224480 + 2.180046706i\)
\(L(\frac12)\) \(\approx\) \(1.333224480 + 2.180046706i\)
\(L(1)\) \(\approx\) \(1.552441480 + 0.5809540673i\)
\(L(1)\) \(\approx\) \(1.552441480 + 0.5809540673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
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 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + T \)
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

−24.68961065874037855714961857698, −24.23986058358440222248094350581, −23.52455356097956761737933840362, −22.45631696389586298569983757362, −21.60974845744269018013147936373, −20.70863565058919356806967796610, −19.9381219894082089753633066528, −19.13826421592252879295279248745, −17.65784033400869168977527304725, −16.498650497842996001620098062098, −15.82334088202695430286716861771, −15.09209584752472035379680404737, −13.53745639160921774469613760514, −13.34379918692849378307495272850, −11.95807115449131793439749963022, −11.437855851431242974476089466290, −10.19214145752553602737545463885, −8.694915473336199091176505556077, −7.79764272055128829866543453984, −6.5441902266487163810893098575, −5.41188803493435871061351064762, −4.53149623073618069944779168788, −3.48659666019775623594706716195, −2.16057563544012050272782167578, −0.51795327102462735692887774425, 1.871422737619894791709731018395, 2.997178940675125889183357528842, 4.0401889715700256702116449776, 5.10228231721443163211648817098, 6.41555739520249756422762694, 7.18997752951925762392391252074, 8.18545901075929867394870046367, 10.03877056487639897969271619119, 10.77145959087505557460010513951, 11.85669481709234617104270857036, 12.571578249753799132968697269149, 13.82600212960847783209543805112, 14.52461031789264687815862936020, 15.54254542581083219191404778181, 16.03230563876739691859694880402, 17.5462877528996545846238403132, 18.49389487438525868044706067975, 19.68257532336708455295967999348, 20.32483040928363536349932735937, 21.43143608793201922230676978974, 22.35464710272098201001300153257, 22.9529246092269243823604370048, 23.75146972269773922553128175046, 24.70918723431972710562683424790, 25.7497767715070156094244412629

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