Properties

Label 1-273-273.125-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: $\chi_{273} (125, \cdot )$
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

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

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