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Properties

Label 2-273-273.272-c1-0-10
Degree $2$
Conductor $273$
Sign $0.466 - 0.884i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.77·2-s + (−1.65 − 0.525i)3-s + 1.15·4-s + 3.85i·5-s + (−2.93 − 0.933i)6-s + (1.56 + 2.12i)7-s − 1.50·8-s + (2.44 + 1.73i)9-s + 6.84i·10-s + 2.15·11-s + (−1.90 − 0.606i)12-s + (−2.48 − 2.61i)13-s + (2.78 + 3.78i)14-s + (2.02 − 6.36i)15-s − 4.97·16-s + 1.53·17-s + ⋯
L(s)  = 1  + 1.25·2-s + (−0.952 − 0.303i)3-s + 0.576·4-s + 1.72i·5-s + (−1.19 − 0.381i)6-s + (0.593 + 0.804i)7-s − 0.531·8-s + (0.815 + 0.578i)9-s + 2.16i·10-s + 0.649·11-s + (−0.549 − 0.175i)12-s + (−0.688 − 0.725i)13-s + (0.745 + 1.01i)14-s + (0.523 − 1.64i)15-s − 1.24·16-s + 0.371·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.466 - 0.884i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47193 + 0.887831i\)
\(L(\frac12)\) \(\approx\) \(1.47193 + 0.887831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 + 0.525i)T \)
7 \( 1 + (-1.56 - 2.12i)T \)
13 \( 1 + (2.48 + 2.61i)T \)
good2 \( 1 - 1.77T + 2T^{2} \)
5 \( 1 - 3.85iT - 5T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 - 0.929iT - 23T^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 - 0.380T + 31T^{2} \)
37 \( 1 + 8.77iT - 37T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 + 1.59iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 - 0.317iT - 59T^{2} \)
61 \( 1 - 1.21iT - 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 7.83T + 71T^{2} \)
73 \( 1 + 1.37T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−11.94081094180545482669573675958, −11.52807799328122029091200558694, −10.60298625552531033678059370786, −9.491052814911975909639647393700, −7.65152008613903862662534507536, −6.82497716647042710322756275760, −5.80324367135118863778584342336, −5.19696258445573177986299337612, −3.66198361851254040252556154855, −2.45948280798052927223975840958, 1.13502723782819019119778788470, 3.89356043169454186075688466080, 4.70254498593602767931970534510, 5.15125351406972904364717861039, 6.30897415745480621451816522657, 7.63254768912424274753362922316, 9.099008993429292050844241872187, 9.800720987797265147476750144511, 11.34658864740288042079128530212, 12.00926246181523851463847176923

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