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Properties

Label 1-273-273.41-r1-0-0
Degree $1$
Conductor $273$
Sign $0.852 - 0.522i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·5-s − i·8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s + i·31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·5-s − i·8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s + i·31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282477415 - 0.3614261739i\)
\(L(\frac12)\) \(\approx\) \(1.282477415 - 0.3614261739i\)
\(L(1)\) \(\approx\) \(0.8063256420 - 0.2110008989i\)
\(L(1)\) \(\approx\) \(0.8063256420 - 0.2110008989i\)

Euler product

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

−25.64965942071834720031067766925, −24.84049240192059502839831334744, −23.9898058706500565281226121952, −22.817905279271053756259842715695, −22.19638258938501537714058099450, −20.79288784759348183493613797329, −19.85183691657615182510881413302, −18.829569484081481090089768965270, −18.36968362984864950080406815246, −17.320527865267943100426604244354, −16.37052381121447978504622362185, −15.532187818459373535099388305011, −14.30461642801769108008188894526, −14.05464164060995936581741980357, −12.046481005810714868583914420507, −11.21270939051572538178767066871, −10.226199506116894351587094173720, −9.39286967242287362562629378780, −8.2288749239726912319432774874, −7.198027690093791811022028858324, −6.419394053535866640866672670, −5.36580425176134717454739025034, −3.58545886978054068359630709594, −2.28925049257603702923624357356, −0.77424037566086484264407401365, 0.90502645176782729644276870263, 1.827543024427916655243258651028, 3.44075990981077973770726093426, 4.53906975251464164187780029132, 6.03526133764313619827392525321, 7.3729266446944454065249759380, 8.30374270161896367396630965668, 9.29305419156054460481871067801, 9.929728201544637887423529519314, 11.29577469593156094030429494644, 12.14734991437298243901703969433, 12.85282393088727691653259175220, 14.10562163935242409421186383679, 15.59315619925057909465707888163, 16.32139385228150077756357440737, 17.40091358136894862809193525766, 17.76383838134717468929030966432, 19.38122411721874237328290090466, 19.682473829429285852466240898942, 20.74767001911591486388480358220, 21.44897131170896309483586040833, 22.487590898406096652223330674, 23.792788969117158197332147249122, 24.75983348201019502266366187587, 25.41803444811034925477286559377

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