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Properties

Label 2-273-91.45-c1-0-6
Degree $2$
Conductor $273$
Sign $0.533 - 0.846i$
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  + (−1.75 + 1.75i)2-s + (−0.866 + 0.5i)3-s − 4.12i·4-s + (−0.286 + 1.06i)5-s + (0.640 − 2.39i)6-s + (1.02 − 2.43i)7-s + (3.72 + 3.72i)8-s + (0.499 − 0.866i)9-s + (−1.37 − 2.37i)10-s + (0.559 − 2.08i)11-s + (2.06 + 3.57i)12-s + (3.47 − 0.970i)13-s + (2.46 + 6.06i)14-s + (−0.286 − 1.06i)15-s − 4.78·16-s + 3.19·17-s + ⋯
L(s)  = 1  + (−1.23 + 1.23i)2-s + (−0.499 + 0.288i)3-s − 2.06i·4-s + (−0.128 + 0.478i)5-s + (0.261 − 0.976i)6-s + (0.388 − 0.921i)7-s + (1.31 + 1.31i)8-s + (0.166 − 0.288i)9-s + (−0.433 − 0.750i)10-s + (0.168 − 0.629i)11-s + (0.595 + 1.03i)12-s + (0.963 − 0.269i)13-s + (0.658 + 1.62i)14-s + (−0.0739 − 0.276i)15-s − 1.19·16-s + 0.774·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.533 - 0.846i$
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: \((2,\ 273,\ (\ :1/2),\ 0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524218 + 0.289260i\)
\(L(\frac12)\) \(\approx\) \(0.524218 + 0.289260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 2.43i)T \)
13 \( 1 + (-3.47 + 0.970i)T \)
good2 \( 1 + (1.75 - 1.75i)T - 2iT^{2} \)
5 \( 1 + (0.286 - 1.06i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.559 + 2.08i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 + (3.43 - 0.920i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.716iT - 23T^{2} \)
29 \( 1 + (-1.58 + 2.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.33 - 0.625i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.98 - 1.98i)T + 37iT^{2} \)
41 \( 1 + (4.01 - 1.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.32 + 4.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 - 2.85i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.54 + 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.76 + 5.76i)T - 59iT^{2} \)
61 \( 1 + (-11.3 - 6.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.4 - 3.07i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.8 + 3.96i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.30 + 4.86i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.63 + 3.63i)T + 83iT^{2} \)
89 \( 1 + (3.84 - 3.84i)T - 89iT^{2} \)
97 \( 1 + (3.61 - 13.4i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.57019361215855302262574745830, −10.63337694048849078656265515092, −10.25807368357471502448101112019, −8.974688939242144746043119844592, −8.125454677470950517745905783842, −7.20714153599420514285986082187, −6.34078175125171030529862610367, −5.42465094966510957266749133160, −3.81654530852978893829548725629, −0.947771085390191469401543707672, 1.21179644724625770967290693785, 2.47604163774271660323630342745, 4.17951904459507656723471331863, 5.67751264344858356253213528455, 7.16666236216509605719674731498, 8.419591968471334239113406490248, 8.844569206012837356692062792416, 9.943429144355741622509912853158, 10.90629941085068470464950880325, 11.58477929998660800324703274343

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