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Properties

Label 2-273-273.269-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.743 + 0.668i$
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  + 2.60i·2-s + (0.206 + 1.71i)3-s − 4.76·4-s + (1.54 + 2.67i)5-s + (−4.47 + 0.537i)6-s + (1.09 − 2.40i)7-s − 7.19i·8-s + (−2.91 + 0.711i)9-s + (−6.96 + 4.02i)10-s + (−2.83 + 1.63i)11-s + (−0.984 − 8.19i)12-s + (3.57 − 0.487i)13-s + (6.25 + 2.85i)14-s + (−4.28 + 3.21i)15-s + 9.17·16-s + 3.84·17-s + ⋯
L(s)  = 1  + 1.83i·2-s + (0.119 + 0.992i)3-s − 2.38·4-s + (0.691 + 1.19i)5-s + (−1.82 + 0.219i)6-s + (0.415 − 0.909i)7-s − 2.54i·8-s + (−0.971 + 0.237i)9-s + (−2.20 + 1.27i)10-s + (−0.854 + 0.493i)11-s + (−0.284 − 2.36i)12-s + (0.990 − 0.135i)13-s + (1.67 + 0.763i)14-s + (−1.10 + 0.829i)15-s + 2.29·16-s + 0.932·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.743 + 0.668i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446172 - 1.16299i\)
\(L(\frac12)\) \(\approx\) \(0.446172 - 1.16299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.206 - 1.71i)T \)
7 \( 1 + (-1.09 + 2.40i)T \)
13 \( 1 + (-3.57 + 0.487i)T \)
good2 \( 1 - 2.60iT - 2T^{2} \)
5 \( 1 + (-1.54 - 2.67i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.83 - 1.63i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.84T + 17T^{2} \)
19 \( 1 + (-2.16 - 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.41iT - 23T^{2} \)
29 \( 1 + (-3.85 - 2.22i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.89 + 3.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.96T + 37T^{2} \)
41 \( 1 + (4.81 - 8.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.70 - 6.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.63 + 2.82i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.67 + 2.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + (8.03 + 4.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.88 - 8.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.88 + 2.24i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.06 - 1.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 + 5.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 - 7.17T + 89T^{2} \)
97 \( 1 + (-10.4 + 6.04i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.09327979833258278910540882927, −11.08189923924256160121626680566, −10.23970562255970984054005935165, −9.659168369021823737375903188939, −8.307569422893172058593375442039, −7.58991511160937500164146482781, −6.50841605468375799953803394018, −5.61970986130302079495993894834, −4.61281257010762610895921625044, −3.33547470820771969928368717257, 1.07422253225149413468250899259, 2.06495504543050942589733704613, 3.31513896217154287414618283214, 5.15053335278366911759557530901, 5.71852461914253214035334457072, 7.982971113111677331881258521066, 8.796500447620864097818949291604, 9.313227682103274206325581509913, 10.62551468226769594212717915262, 11.57254837646526417289472136100

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