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Properties

Label 2-273-91.5-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.879 + 0.475i$
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  + (−0.677 − 2.52i)2-s + (−0.866 + 0.5i)3-s + (−4.20 + 2.42i)4-s + (3.92 − 1.05i)5-s + (1.85 + 1.85i)6-s + (1.12 − 2.39i)7-s + (5.28 + 5.28i)8-s + (0.499 − 0.866i)9-s + (−5.31 − 9.20i)10-s + (−0.150 + 0.560i)11-s + (2.42 − 4.20i)12-s + (2.59 − 2.50i)13-s + (−6.81 − 1.21i)14-s + (−2.87 + 2.87i)15-s + (4.93 − 8.54i)16-s + (−0.229 − 0.397i)17-s + ⋯
L(s)  = 1  + (−0.479 − 1.78i)2-s + (−0.499 + 0.288i)3-s + (−2.10 + 1.21i)4-s + (1.75 − 0.469i)5-s + (0.755 + 0.755i)6-s + (0.424 − 0.905i)7-s + (1.86 + 1.86i)8-s + (0.166 − 0.288i)9-s + (−1.68 − 2.91i)10-s + (−0.0453 + 0.169i)11-s + (0.700 − 1.21i)12-s + (0.720 − 0.693i)13-s + (−1.82 − 0.325i)14-s + (−0.741 + 0.741i)15-s + (1.23 − 2.13i)16-s + (−0.0556 − 0.0964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248668 - 0.983703i\)
\(L(\frac12)\) \(\approx\) \(0.248668 - 0.983703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 + 2.39i)T \)
13 \( 1 + (-2.59 + 2.50i)T \)
good2 \( 1 + (0.677 + 2.52i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-3.92 + 1.05i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.150 - 0.560i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.229 + 0.397i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.44 - 1.99i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0405 - 0.0233i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.08T + 29T^{2} \)
31 \( 1 + (-1.31 + 4.88i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.23 - 1.13i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.50 - 4.50i)T + 41iT^{2} \)
43 \( 1 + 0.884iT - 43T^{2} \)
47 \( 1 + (-0.852 - 3.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.253 - 0.438i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.15 - 0.577i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.165 - 0.0958i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.62 - 2.31i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.23 - 4.23i)T - 71iT^{2} \)
73 \( 1 + (6.99 + 1.87i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.92 + 5.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.67 - 9.67i)T + 83iT^{2} \)
89 \( 1 + (-4.88 - 18.2i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.63 - 5.63i)T + 97iT^{2} \)
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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

−11.12925949445116620469279858655, −10.51173882789347590496823909045, −10.00718717907669909260766449135, −9.117317597484159309139387209125, −8.169433500868535968070877958284, −6.23938044455924111035357957027, −4.99513998720050540331287330115, −3.95253719030502276273604319313, −2.25465784346324263809386050635, −1.09968783871498682305825502243, 1.92035005897184531961704098410, 4.80906757723958774383525202300, 5.80166153006499400669585887006, 6.26781294105733640598138482845, 7.04287859178861767841394320727, 8.593886134847031629114150161068, 9.005940449528294746826496858093, 10.13160809531314264811037213790, 11.02005671997007216313288419655, 12.70188693516366627035722679647

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