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Properties

Label 2-273-91.20-c1-0-10
Degree $2$
Conductor $273$
Sign $-0.558 + 0.829i$
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 + (0.347 + 2.62i)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 + (0.131 + 0.991i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.558 + 0.829i$
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: \((2,\ 273,\ (\ :1/2),\ -0.558 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0871850 - 0.163911i\)
\(L(\frac12)\) \(\approx\) \(0.0871850 - 0.163911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.347 - 2.62i)T \)
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

−11.50350776371942349837523499437, −10.58106302934807826080276978722, −9.582391063709861603776749726104, −8.520298471949375370547991408752, −7.931000142285550528504133892560, −6.74873567592229457488033865833, −5.63943592346103304329353382028, −4.38245780182971873120588490719, −2.57747087821286295498276556548, −0.19440537909953054507377038390, 1.74426828515339886888506195661, 4.26538912963582212245352832886, 4.73153435954456747455437156313, 6.39228522735382235720102488336, 7.55952639758294509163614110038, 8.477716847778862414541945151469, 9.599633650801532305389920685640, 10.33693613579736985339897438925, 10.88422017247804380532919597513, 12.11293690707118123766227265869

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