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Properties

Label 2-273-91.25-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.999 - 0.0344i$
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  + (−0.504 + 0.291i)2-s + (0.5 − 0.866i)3-s + (−0.830 + 1.43i)4-s + (−1.26 + 0.728i)5-s + 0.582i·6-s + (−2.46 + 0.952i)7-s − 2.13i·8-s + (−0.499 − 0.866i)9-s + (0.424 − 0.735i)10-s + (−2.68 − 1.54i)11-s + (0.830 + 1.43i)12-s + (−3.54 + 0.660i)13-s + (0.968 − 1.19i)14-s + 1.45i·15-s + (−1.03 − 1.80i)16-s + (1.75 − 3.03i)17-s + ⋯
L(s)  = 1  + (−0.356 + 0.205i)2-s + (0.288 − 0.499i)3-s + (−0.415 + 0.719i)4-s + (−0.564 + 0.325i)5-s + 0.237i·6-s + (−0.932 + 0.359i)7-s − 0.754i·8-s + (−0.166 − 0.288i)9-s + (0.134 − 0.232i)10-s + (−0.808 − 0.466i)11-s + (0.239 + 0.415i)12-s + (−0.983 + 0.183i)13-s + (0.258 − 0.320i)14-s + 0.376i·15-s + (−0.259 − 0.450i)16-s + (0.424 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.999 - 0.0344i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.999 - 0.0344i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00295253 + 0.171387i\)
\(L(\frac12)\) \(\approx\) \(0.00295253 + 0.171387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.46 - 0.952i)T \)
13 \( 1 + (3.54 - 0.660i)T \)
good2 \( 1 + (0.504 - 0.291i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.26 - 0.728i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.68 + 1.54i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.75 + 3.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.96 - 2.28i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.01 - 5.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.81T + 29T^{2} \)
31 \( 1 + (-7.82 - 4.52i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.72 - 2.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.30iT - 41T^{2} \)
43 \( 1 + 8.12T + 43T^{2} \)
47 \( 1 + (-9.47 + 5.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.62 - 6.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.86 + 2.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.83 - 6.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 - 6.31i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.67iT - 71T^{2} \)
73 \( 1 + (1.45 + 0.842i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.48iT - 83T^{2} \)
89 \( 1 + (13.1 - 7.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.56iT - 97T^{2} \)
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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

−12.36234951998673203768271608998, −11.73264939892423031716082041822, −10.26990927286763437938665319797, −9.332316463866895323253648971033, −8.446232793324971945114180971742, −7.48333469728204501177148830732, −6.90199997574174530605448691075, −5.37071314027481834551542264156, −3.68535127577709598726753089771, −2.78827469068904308508432167909, 0.13460339606015733152637379113, 2.53732330226061284962975848492, 4.17864313749184171770885538403, 5.04370240847303491297427788191, 6.40828689456233664758173209198, 7.80865235557103233485934721938, 8.682094258194713069232219189536, 9.796333158214864165254652120977, 10.19151688795233524356071868243, 11.12564734898596095936156909696

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