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Properties

Label 2-273-91.45-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.942 - 0.334i$
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  + (−1.59 + 1.59i)2-s + (−0.866 + 0.5i)3-s − 3.10i·4-s + (0.609 − 2.27i)5-s + (0.584 − 2.18i)6-s + (1.40 + 2.23i)7-s + (1.76 + 1.76i)8-s + (0.499 − 0.866i)9-s + (2.65 + 4.60i)10-s + (−1.30 + 4.87i)11-s + (1.55 + 2.68i)12-s + (−1.62 + 3.21i)13-s + (−5.82 − 1.32i)14-s + (0.609 + 2.27i)15-s + 0.576·16-s − 5.28·17-s + ⋯
L(s)  = 1  + (−1.12 + 1.12i)2-s + (−0.499 + 0.288i)3-s − 1.55i·4-s + (0.272 − 1.01i)5-s + (0.238 − 0.890i)6-s + (0.532 + 0.846i)7-s + (0.623 + 0.623i)8-s + (0.166 − 0.288i)9-s + (0.841 + 1.45i)10-s + (−0.394 + 1.47i)11-s + (0.447 + 0.775i)12-s + (−0.450 + 0.892i)13-s + (−1.55 − 0.354i)14-s + (0.157 + 0.587i)15-s + 0.144·16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.942 - 0.334i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.942 - 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0836832 + 0.485882i\)
\(L(\frac12)\) \(\approx\) \(0.0836832 + 0.485882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 2.23i)T \)
13 \( 1 + (1.62 - 3.21i)T \)
good2 \( 1 + (1.59 - 1.59i)T - 2iT^{2} \)
5 \( 1 + (-0.609 + 2.27i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.30 - 4.87i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + (-4.08 + 1.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.98iT - 23T^{2} \)
29 \( 1 + (-0.565 + 0.978i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.23 - 1.40i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.75 - 3.75i)T + 37iT^{2} \)
41 \( 1 + (5.61 - 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.65 - 5.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.28 - 1.95i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.538 - 0.931i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.97 + 5.97i)T - 59iT^{2} \)
61 \( 1 + (2.73 + 1.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.13 - 1.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12.9 - 3.47i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.418 - 1.56i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.31 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.21 - 7.21i)T + 83iT^{2} \)
89 \( 1 + (-6.32 + 6.32i)T - 89iT^{2} \)
97 \( 1 + (-4.32 + 16.1i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.19731917061848505412766185555, −11.30792003148284697891462746914, −9.828924465998931222700758760659, −9.375231329604096108467516663429, −8.607184518519700600031440815554, −7.49477067257521653764945298320, −6.58041364169824601643507366294, −5.27879643363671356679443382967, −4.74810590846816835296700945424, −1.75009742569453035371672851797, 0.58171983299135831985065702486, 2.34290834633031978438708748448, 3.50778084897268973244620244374, 5.42653179423741756918601796773, 6.79736827119755792019940880738, 7.79542595366148332875087291278, 8.677050272154996095033063420830, 10.01886301371128085618035383226, 10.81271574785847293108290651538, 10.94952687415475923406699208814

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