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Properties

Label 2-273-273.191-c0-0-0
Degree $2$
Conductor $273$
Sign $-0.313 - 0.949i$
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)2-s + i·3-s + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)6-s + 7-s − i·8-s − 9-s − 0.999·10-s − 13-s + (−0.866 + 0.5i)14-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + i·21-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + i·3-s + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)6-s + 7-s − i·8-s − 9-s − 0.999·10-s − 13-s + (−0.866 + 0.5i)14-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + i·21-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.313 - 0.949i$
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: \((2,\ 273,\ (\ :0),\ -0.313 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5508547248\)
\(L(\frac12)\) \(\approx\) \(0.5508547248\)
\(L(1)\) 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 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−12.24573181948763386845513585970, −11.04501307154811465532805628186, −10.33626314284726457354710881012, −9.430921381599960827968733502190, −8.820357422139587213868686433031, −7.71666080269870697623348431477, −6.66648145194786133686582957984, −5.33144545212742784641430491873, −4.26645754761237245793452582213, −2.55379765570476334429151299596, 1.46701386008400956752707974800, 2.34682277632814648869917901186, 4.93211054895975106987497877462, 5.74769694381576099472239387629, 7.19659665948944051596930199674, 8.217456342285713154561538748599, 8.977827777464243947026480799234, 9.818872203035465549130655288651, 11.01329888553444399217411246081, 11.60609477133265737722279990486

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