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Properties

Label 2-273-273.191-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9983823539\)
\(L(\frac12)\) \(\approx\) \(0.9983823539\)
\(L(1)\) 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

−12.05737747511877156880135756866, −11.67679062703311331743736437548, −10.43893284793884885697851651955, −8.586813980643473255851884913155, −8.102510253783986063133748923304, −7.22015911817377249092855803164, −5.54177515987377050736600987747, −4.69790133843077734416287782185, −3.43217743974993885628629774674, −1.91498777277529916463850108324, 3.09985039398409065577220373210, 4.35513678342534857090569298681, 4.90190141434814279554660513346, 6.06171164672452308306480936993, 7.41491737937869675637560611374, 8.277139040614089191464118512021, 9.733548646959458786139553850933, 10.35910340508297083824188907895, 11.61821887565430342132888060443, 12.04003221924666949494666668799

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