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Properties

Label 2-273-91.20-c1-0-7
Degree $2$
Conductor $273$
Sign $0.981 - 0.189i$
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  + (−1.11 + 0.299i)2-s + (0.866 + 0.5i)3-s + (−0.576 + 0.332i)4-s + (0.549 − 0.549i)5-s + (−1.11 − 0.299i)6-s + (1.61 − 2.09i)7-s + (2.17 − 2.17i)8-s + (0.499 + 0.866i)9-s + (−0.448 + 0.777i)10-s + (−0.824 − 3.07i)11-s − 0.665·12-s + (2.63 + 2.45i)13-s + (−1.17 + 2.82i)14-s + (0.750 − 0.201i)15-s + (−1.11 + 1.92i)16-s + (1.74 + 3.03i)17-s + ⋯
L(s)  = 1  + (−0.789 + 0.211i)2-s + (0.499 + 0.288i)3-s + (−0.288 + 0.166i)4-s + (0.245 − 0.245i)5-s + (−0.455 − 0.122i)6-s + (0.609 − 0.792i)7-s + (0.769 − 0.769i)8-s + (0.166 + 0.288i)9-s + (−0.141 + 0.245i)10-s + (−0.248 − 0.927i)11-s − 0.192·12-s + (0.731 + 0.682i)13-s + (−0.313 + 0.754i)14-s + (0.193 − 0.0519i)15-s + (−0.278 + 0.482i)16-s + (0.424 + 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02897 + 0.0984891i\)
\(L(\frac12)\) \(\approx\) \(1.02897 + 0.0984891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-1.61 + 2.09i)T \)
13 \( 1 + (-2.63 - 2.45i)T \)
good2 \( 1 + (1.11 - 0.299i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.549 + 0.549i)T - 5iT^{2} \)
11 \( 1 + (0.824 + 3.07i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.74 - 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.06 - 1.62i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.54 + 7.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.888 - 0.888i)T - 31iT^{2} \)
37 \( 1 + (-0.151 - 0.564i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.704 - 2.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.60 + 3.81i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.267 - 0.267i)T + 47iT^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-0.635 + 2.37i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.70 - 3.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.90 - 1.85i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.51 - 9.37i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.71 + 7.71i)T + 73iT^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \)
89 \( 1 + (7.02 - 1.88i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.704 + 0.188i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.75323510281423356000252991748, −10.65268283390300377278427342150, −9.906086220723881019592480830586, −8.958870433824898857326090964454, −8.144718812432671524018915469076, −7.53840905219684043020700048105, −6.00295054600695001435464374555, −4.50029310172432931373419796766, −3.54921720450442896148277548155, −1.29819554133428281029831170632, 1.48584273990800747677525045635, 2.85350704356312751341178686140, 4.75947144015638803452407448338, 5.77302280047529624409925204392, 7.41644997085079919369822548139, 8.108523604644904527709650672642, 9.096088111324768081271494153269, 9.787496765930312168235462525485, 10.71273418920296543674305469316, 11.79188984169133411636386675410

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