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Properties

Label 2-273-91.33-c1-0-12
Degree $2$
Conductor $273$
Sign $0.926 + 0.375i$
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  + (−0.449 + 1.67i)2-s − i·3-s + (−0.885 − 0.511i)4-s + (−0.524 − 1.95i)5-s + (1.67 + 0.449i)6-s + (−0.616 − 2.57i)7-s + (−1.20 + 1.20i)8-s − 9-s + 3.52·10-s + (2.56 − 2.56i)11-s + (−0.511 + 0.885i)12-s + (1.78 − 3.13i)13-s + (4.59 + 0.123i)14-s + (−1.95 + 0.524i)15-s + (−2.49 − 4.32i)16-s + (−0.752 + 1.30i)17-s + ⋯
L(s)  = 1  + (−0.318 + 1.18i)2-s − 0.577i·3-s + (−0.442 − 0.255i)4-s + (−0.234 − 0.875i)5-s + (0.685 + 0.183i)6-s + (−0.232 − 0.972i)7-s + (−0.424 + 0.424i)8-s − 0.333·9-s + 1.11·10-s + (0.772 − 0.772i)11-s + (−0.147 + 0.255i)12-s + (0.496 − 0.868i)13-s + (1.22 + 0.0328i)14-s + (−0.505 + 0.135i)15-s + (−0.624 − 1.08i)16-s + (−0.182 + 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.969391 - 0.188908i\)
\(L(\frac12)\) \(\approx\) \(0.969391 - 0.188908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (0.616 + 2.57i)T \)
13 \( 1 + (-1.78 + 3.13i)T \)
good2 \( 1 + (0.449 - 1.67i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.524 + 1.95i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.56 + 2.56i)T - 11iT^{2} \)
17 \( 1 + (0.752 - 1.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 + 2.92i)T - 19iT^{2} \)
23 \( 1 + (3.21 - 1.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.84 - 3.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.05 - 1.89i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.52 + 0.677i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.33 - 4.97i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.51 - 2.60i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.255 + 0.0684i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.63 + 4.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.24 + 0.334i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + (-9.48 - 9.48i)T + 67iT^{2} \)
71 \( 1 + (-3.11 + 11.6i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.744 - 2.77i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.09 - 14.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.3 + 10.3i)T - 83iT^{2} \)
89 \( 1 + (2.60 - 9.72i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.645 - 0.173i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.89918737588625927957776486410, −11.00801277688028892762760285591, −9.580024402617244368114916646074, −8.491634024372078585358894967789, −8.016857058462430211238602451340, −6.91842377980054855161569263612, −6.15771949178571475533378942543, −4.98366847189238389982577775682, −3.39524890690245044100651432202, −0.883303884580761441661715062658, 2.03267322909220907518490370854, 3.19744428621388232543840758845, 4.24509649495812454669450699512, 6.00307881404463479033100567192, 6.92903267376326334666240172698, 8.578417579458423588611181894660, 9.496922000224253904713938359675, 10.05420268451923574153179089605, 11.10125332715105574808030856170, 11.81828431502505724387464529013

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