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Properties

Label 1-273-273.128-r0-0-0
Degree $1$
Conductor $273$
Sign $0.101 - 0.994i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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 + (−0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 16-s + 17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + 16-s + 17-s + (−0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.101 - 0.994i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.101 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7107305230 - 0.6417039896i\)
\(L(\frac12)\) \(\approx\) \(0.7107305230 - 0.6417039896i\)
\(L(1)\) \(\approx\) \(0.7924940068 - 0.4118404395i\)
\(L(1)\) \(\approx\) \(0.7924940068 - 0.4118404395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + iT \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + iT \)
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

−25.59577539565116444057000174347, −25.1479692286369072531726313748, −24.04740529817811034624703043606, −23.351399960804606399522335644, −22.67537580697975674591480182539, −21.552506248762054110320076267, −20.38559021379346014124469200527, −19.3102206080964389989510234641, −18.617898638228491395148297560039, −17.24484826361898936826297020434, −16.73556886444408047326238865765, −15.75326800458616369506264023204, −14.87299846269819511980366840670, −14.14295352656073491364435014915, −12.70610025774601192270802950713, −12.2042027018615107712225327573, −10.70615375009671644654321176095, −9.36474760957036824736352461827, −8.59030340283182704745471685197, −7.574527816024705334536354920194, −6.72925450611900728165810506662, −5.42603577922107471285497202291, −4.41126911283141243519146895720, −3.49154088050765919636596253836, −1.16071415076833191952673519279, 0.86230501687707515125956139783, 2.53569679149283701876700826107, 3.59871905808262628504924160925, 4.42137364277459741276704354770, 5.90100721424199888150211295822, 7.31617398778363814056318295253, 8.43269028820453350701932126275, 9.37318192822661763347108003293, 10.61300593221580174412903862602, 11.32942982324708886455838524734, 12.13161367205411736168563512186, 13.127839816918932378489099488107, 14.34305058654056967092325189282, 14.961580876166651542646267083826, 16.39616302416573217426092562834, 17.36676441269970833571508554322, 18.52044775653810748097508133108, 19.29208211240067793827418058328, 19.75780163828823876328462437243, 21.01907354814892341962728412087, 21.74448054029597305741291819221, 22.811158170920041495705533994648, 23.286865267601523763557713502231, 24.41672803878134242441388814111, 25.75339648369434676967047460899

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