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Properties

Label 2-273-91.54-c1-0-14
Degree $2$
Conductor $273$
Sign $0.947 + 0.319i$
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  + (0.884 − 0.884i)2-s + (0.866 + 0.5i)3-s + 0.434i·4-s + (3.68 − 0.986i)5-s + (1.20 − 0.323i)6-s + (−2.53 − 0.767i)7-s + (2.15 + 2.15i)8-s + (0.499 + 0.866i)9-s + (2.38 − 4.12i)10-s + (−4.32 + 1.15i)11-s + (−0.217 + 0.376i)12-s + (−3.57 + 0.437i)13-s + (−2.91 + 1.56i)14-s + (3.68 + 0.986i)15-s + 2.94·16-s + 1.20·17-s + ⋯
L(s)  = 1  + (0.625 − 0.625i)2-s + (0.499 + 0.288i)3-s + 0.217i·4-s + (1.64 − 0.441i)5-s + (0.493 − 0.132i)6-s + (−0.957 − 0.289i)7-s + (0.761 + 0.761i)8-s + (0.166 + 0.288i)9-s + (0.753 − 1.30i)10-s + (−1.30 + 0.349i)11-s + (−0.0627 + 0.108i)12-s + (−0.992 + 0.121i)13-s + (−0.780 + 0.417i)14-s + (0.950 + 0.254i)15-s + 0.735·16-s + 0.291·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19482 - 0.359759i\)
\(L(\frac12)\) \(\approx\) \(2.19482 - 0.359759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 0.767i)T \)
13 \( 1 + (3.57 - 0.437i)T \)
good2 \( 1 + (-0.884 + 0.884i)T - 2iT^{2} \)
5 \( 1 + (-3.68 + 0.986i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.32 - 1.15i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
19 \( 1 + (-1.38 + 5.18i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.38iT - 23T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.38 - 5.16i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.94 - 4.94i)T + 37iT^{2} \)
41 \( 1 + (0.914 - 3.41i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.74 + 5.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 - 4.98i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.14 + 1.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.30 - 4.30i)T - 59iT^{2} \)
61 \( 1 + (-4.68 + 2.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.40 + 5.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.05 - 11.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.16 - 1.92i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.76 + 9.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.00 - 5.00i)T + 83iT^{2} \)
89 \( 1 + (3.72 - 3.72i)T - 89iT^{2} \)
97 \( 1 + (2.69 - 0.722i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.20410759681226480344160198195, −10.73060005199113542348742196817, −9.939419729017799462250694556632, −9.378784262614622061738088425106, −8.070993877632304839037455185942, −6.85690553733797325698738557401, −5.38564171308670791005980539959, −4.62629798537090236465813364299, −2.95068977515542190924184146838, −2.26354421448092815010146318011, 2.05015542594606826538107480571, 3.28708848457427402095554056324, 5.35811340906183167520116361031, 5.76667647495226383996432664537, 6.80382909685946087754769134449, 7.73914860723192809934759536372, 9.434073739746269326722868625774, 9.878169575432276187924585336059, 10.65746459590924420319655860569, 12.53413977231874114925631991431

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