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Properties

Label 1-273-273.5-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.348 - 0.937i$
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 + (−0.866 + 0.5i)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 + i·20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − 29-s + (−0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)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 + i·20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − 29-s + (−0.866 − 0.5i)31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454247137 - 2.093058862i\)
\(L(\frac12)\) \(\approx\) \(1.454247137 - 2.093058862i\)
\(L(1)\) \(\approx\) \(1.411938446 - 0.6740996381i\)
\(L(1)\) \(\approx\) \(1.411938446 - 0.6740996381i\)

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 \( 1 + (-0.866 + 0.5i)T \)
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 \( 1 - T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + iT \)
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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

−25.50238301697962945686910617767, −24.65996926364321613841517382729, −23.88560221172559941332931258756, −23.22858281714879676880167020356, −22.198208825523253970718414628685, −21.4597410253234850770321247570, −20.27823582638515295922445359475, −19.69667359242229994255969630388, −18.41493538965173015880691503403, −16.95198911220197117418616562237, −16.51740686178269736367632690685, −15.48005083199891927663172539255, −14.66889714325206079430468240314, −13.69771800748445101717459831676, −12.6242163706334891142541989236, −11.85982967175317809896941705502, −11.07105560068593007712964343843, −9.348285919675821343010134272172, −8.18498553483109022641514398895, −7.47731349711212030415248555460, −6.19450053530742172546015614338, −5.22815184309601377616695461512, −3.96425216472325039928712624278, −3.36765586785877650537667968564, −1.434963316334812713947708747875, 0.61341549491801727622292576147, 2.23038232316506599691782105988, 3.44594625284467310455066416445, 4.261963699493745736501530455600, 5.45803623277626925192808556447, 6.77889118982676174600961371606, 7.506085893482914197377644270186, 9.184467409674709870984538930312, 10.22579876524987818019483795241, 11.400485505213053071550403335979, 11.86783167984822139455697800044, 12.908096797429185039316976557710, 14.15933064390296467457221847964, 14.746856316994906898752687193162, 15.70851077807211546785556181893, 16.606725855052670793038974951733, 18.22431960889808421305320240432, 18.93039868994649423367824174032, 20.09472339517230657456363155752, 20.36683177810678831957337493911, 21.86955470130160294896794398691, 22.48299952292543696718392005700, 23.163731242496370124709918241434, 24.133717861290278330742337108960, 24.92306629835473557970238616735

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