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Properties

Label 1-273-273.152-r0-0-0
Degree $1$
Conductor $273$
Sign $-0.923 + 0.384i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 8-s − 10-s − 11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 8-s − 10-s − 11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.923 + 0.384i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ -0.923 + 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1538329812 - 0.7688240641i\)
\(L(\frac12)\) \(\approx\) \(-0.1538329812 - 0.7688240641i\)
\(L(1)\) \(\approx\) \(0.6199163425 - 0.6760465630i\)
\(L(1)\) \(\approx\) \(0.6199163425 - 0.6760465630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−26.15286476508305525412105055663, −25.40319993508379456330497338673, −24.27600348010669469323446219673, −23.31112572366721071209888480801, −23.016430619900199697300574382652, −21.69684636987684906382129561828, −21.24594644763552050414750349666, −19.70179301644425951780164199696, −18.74570317443158475547386865087, −17.82088782316767305311534505176, −16.97331794025324931936796784301, −15.57841037822610562958866808308, −15.37005324956234389639737754498, −14.25645075604565088548568185946, −13.30590708621297449265483675104, −12.37548284506118523390039477169, −11.18828236316703544211266052803, −10.19279116217901048063317046461, −8.64918860424676295345566534064, −7.82295149847264593576285231067, −6.86404059159864729033971730176, −5.954462647929405209768817769582, −4.68682956915658742245344761548, −3.61687111062895719134676004037, −2.51296427003893561555188962036, 0.4312370260005504886307116007, 2.03715758638410575712474375669, 3.25465107400617044566246968481, 4.596076837696889436968329355553, 5.118166004082213676210914928603, 6.58803946534783862428997408330, 8.165073831487431310139546324313, 9.01001434741334097090008408160, 10.19147018692020015987057103564, 11.13369031865254376461908855686, 12.11535634407036508350107488518, 12.929986378178358725978716480822, 13.63481663105686607639144688712, 14.96278243713967430557370522929, 15.7474090207202568221380412873, 16.86874933901525733472834623432, 18.14041673556922870921372023722, 18.97050648995320802803768148437, 19.94308430174068455057980809823, 20.68815081984803815991116239165, 21.31165678695454638294635974297, 22.51851917976650262681925743946, 23.33546232767277834411965279267, 24.05097314058928674544582414392, 24.896188698399007878204682311726

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