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Properties

Label 2-273-273.206-c0-0-0
Degree $2$
Conductor $273$
Sign $0.635 + 0.771i$
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  − i·3-s + (0.866 − 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + (−0.366 + 0.366i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)25-s + i·27-s + 0.999i·28-s + (−0.366 − 1.36i)31-s + (−0.866 + 0.5i)36-s + (0.5 + 1.86i)37-s + ⋯
L(s)  = 1  − i·3-s + (0.866 − 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.499 − 0.866i)16-s + (−0.366 + 0.366i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)25-s + i·27-s + 0.999i·28-s + (−0.366 − 1.36i)31-s + (−0.866 + 0.5i)36-s + (0.5 + 1.86i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.635 + 0.771i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8191014848\)
\(L(\frac12)\) \(\approx\) \(0.8191014848\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 - iT - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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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

−11.96058346482206502516531793266, −11.38638383623674708729858974309, −10.13351917436779899710432283742, −9.128962960172365827268153908458, −7.954794161978453483082804853458, −6.89886168472651957365173336183, −6.23209285646454801820254691428, −5.20448120390647946006055764545, −2.95496702240470255425511401749, −1.90838659093179909856543945911, 2.73138368709904615285565722032, 3.75303998144454337047904639610, 4.99547877614419799826341225903, 6.40346037199099792192881583171, 7.34422395005505103383706224489, 8.430785080488699196297846898803, 9.616580360163929324624946920287, 10.53110970486695897649286361158, 11.04032795834345718993695084373, 12.21970123440303917988899681682

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