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Properties

Label 2-273-273.191-c0-0-1
Degree $2$
Conductor $273$
Sign $0.949 - 0.313i$
Analytic cond. $0.136244$
Root an. cond. $0.369113$
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  + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + 9-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 27-s + 0.999·28-s + (−1 − 1.73i)31-s + (−0.5 + 0.866i)36-s + (0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)7-s + 9-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 27-s + 0.999·28-s + (−1 − 1.73i)31-s + (−0.5 + 0.866i)36-s + (0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8716818788\)
\(L(\frac12)\) \(\approx\) \(0.8716818788\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.53134817347494461545917938812, −11.28312144833772637619063067850, −9.927840296809085645880369734637, −9.345112294070437519236864962480, −8.269488523470644085255250662983, −7.51989052799530073639003036007, −6.56356817623343802398847747667, −4.43874307070481477132483183021, −3.85365541895090556900335790972, −2.46618336311894048396802786649, 2.08137434854167615783808386896, 3.49716115974458044202166729864, 4.94176371987835500257443711914, 5.98868748900753155231223910514, 7.27257749214163409643245943034, 8.554337223331167530883043376707, 9.167265071928771324305868643822, 10.00632972835089425287224747310, 10.85357918957215016724031437838, 12.47280104986919417425236328904

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