Properties

Label 2-273-91.41-c1-0-6
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: $\chi_{273} (223, \cdot )$
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

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

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