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Properties

Label 1-273-273.59-r1-0-0
Degree $1$
Conductor $273$
Sign $0.931 - 0.362i$
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 + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.931 - 0.362i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.931 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7754945332 - 0.1455506885i\)
\(L(\frac12)\) \(\approx\) \(0.7754945332 - 0.1455506885i\)
\(L(1)\) \(\approx\) \(0.6293141631 - 0.3395415936i\)
\(L(1)\) \(\approx\) \(0.6293141631 - 0.3395415936i\)

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

−25.68568425064696055877359976154, −24.54325639381457856169454267229, −23.68600852791192979728154353151, −23.055240660359563643180566992817, −22.25619471069483904423763959447, −21.164173411593377316918643003593, −19.82553375035822121421586465758, −18.93894467561208807643973623217, −18.10027637359264069747866770837, −17.21449384609231705661571211004, −16.07598376304677219612008868246, −15.285612099493965970686758303024, −14.79748909179351866969788547560, −13.4348198513080522539962599634, −12.69339747401635579870172525180, −11.305429604706831573954771130620, −10.305103355705523333297509034504, −9.023903162913477914738732448762, −8.04215313023448854280985628904, −7.196574529357040579096299366361, −6.330185539836268578123131408824, −4.904188171468875253241031879119, −4.09138862500920968463255462747, −2.63748572402811903098509335979, −0.365099683096231204695746554458, 0.78428597902106022152610446420, 2.32639209638966839841679347823, 3.55581777194456291415610084679, 4.530047138634501930478734877993, 5.54790997110994000143148281867, 7.32464668921984999888269266102, 8.50132219481796349355355669913, 9.07530978276759813634186775534, 10.686532477386080624092741444703, 11.054788455178144215005846880010, 12.41318566184063163877657660088, 12.86707049135461707135335171365, 14.020752844477107491210321204146, 15.19941078792704212991869011808, 16.21068692899313206566154192066, 17.28437072399372759195613642897, 18.37737730567241552967052686960, 19.19777747419815073481538049103, 19.939980875611256660573159557620, 20.838117105862193899371051878985, 21.56888599153690307687906585963, 22.70645752861919287065080380088, 23.50856513513089402393185438341, 24.18227123514275738024196101254, 25.57277891711903410278162699224

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