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Properties

Label 1-273-273.83-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.957 - 0.289i$
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  + i·2-s − 4-s − i·5-s − i·8-s + 10-s − i·11-s + 16-s − 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s − 29-s + i·31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·5-s − i·8-s + 10-s − i·11-s + 16-s − 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s − 29-s + i·31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02382526336 + 0.1609067754i\)
\(L(\frac12)\) \(\approx\) \(0.02382526336 + 0.1609067754i\)
\(L(1)\) \(\approx\) \(0.7279176310 + 0.2203957249i\)
\(L(1)\) \(\approx\) \(0.7279176310 + 0.2203957249i\)

Euler product

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

−25.14295520230212960257645602345, −23.75474302843969784596468184846, −22.86285330152255481756593185469, −22.23718991797603342922573152662, −21.4155993976256111813533452380, −20.313070273186591936177058530986, −19.60056403044861707023292786125, −18.607958248905475607226322634742, −17.89369472931817573976115407229, −17.08455545462901349210403808406, −15.334411897033557744003700225092, −14.7344561854325021444596084271, −13.52640225136290567147436716885, −12.802836450044202847491172567352, −11.4797141617373661854233844141, −10.96220971992675853607455126832, −9.86600421823342223432949815237, −9.063639554484638609691632064388, −7.63562636151697961618480186951, −6.5931914971002069695431834105, −5.0729085561195520836973562354, −3.99851549158183118386954870495, −2.79326125033551321357181938996, −1.896003167415264962478116784823, −0.05230306575999167489188760915, 1.321310702307152130356659560457, 3.45937081918149033217150622572, 4.63803969999198709349943608809, 5.54269628248608801508828876273, 6.53198342878171541275833661373, 7.83015757518690776364432203410, 8.67879826021385943341259351508, 9.36969026470641986659225403767, 10.75781044901782137691608222776, 12.15444585583765830944463380651, 13.15496503508294944517184413481, 13.822311254289219815171801524107, 15.00154285378326264972459518355, 15.96137493573433953343484462595, 16.672794734679533381579087922316, 17.388240118630382632416417671083, 18.5354516714829838071962547813, 19.40161490192325601573450712580, 20.61581082121237451264689421171, 21.560363202167245146901362883405, 22.51745037979409859187800203621, 23.550830297380229706167683411335, 24.31291832360959219782968431928, 24.880400419873621399933691037508, 25.813400827508082578814768997419

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