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Properties

Label 2-273-273.236-c0-0-0
Degree $2$
Conductor $273$
Sign $0.988 + 0.151i$
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  + (0.866 + 0.5i)3-s − i·4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s − 16-s + (−0.5 + 1.86i)19-s + (−0.499 − 0.866i)21-s + (0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 + 0.866i)28-s + (0.366 − 1.36i)31-s + (0.866 − 0.499i)36-s + (−0.366 − 0.366i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s − i·4-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s − 16-s + (−0.5 + 1.86i)19-s + (−0.499 − 0.866i)21-s + (0.866 − 0.5i)25-s + 0.999i·27-s + (−0.5 + 0.866i)28-s + (0.366 − 1.36i)31-s + (0.866 − 0.499i)36-s + (−0.366 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.988 + 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8841725027\)
\(L(\frac12)\) \(\approx\) \(0.8841725027\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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 \( 1 + iT^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−12.18632022449698427180098031058, −10.76252135968319292679564830951, −10.02169589208203451454937177040, −9.572610669982683841410008885075, −8.421069920951592564501845676259, −7.19970698996099435722616349042, −6.13904334681480548249363222274, −4.79212920581174562751973967591, −3.69437790266692302866185565657, −2.12255267909996772894638217098, 2.59829944923730075654488430825, 3.26023046269095444679245470634, 4.81506723547575448029043494977, 6.67667278842671113982667147721, 7.19999517213636921111326628875, 8.458275221751799179445986034992, 8.995853668757567558734547247849, 10.02778342373682153317933010482, 11.50039269326311012155975661685, 12.48118834808656393411396723268

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