# Properties

 Label 1-273-273.23-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 $0$

# Related objects

## 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: $\chi_{273} (23, \cdot )$ 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

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