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Properties

Label 2-273-91.51-c1-0-10
Degree $2$
Conductor $273$
Sign $0.533 - 0.845i$
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  + (2.25 + 1.30i)2-s + (−0.5 − 0.866i)3-s + (2.39 + 4.14i)4-s + (−0.401 − 0.232i)5-s − 2.60i·6-s + (2.25 − 1.38i)7-s + 7.25i·8-s + (−0.499 + 0.866i)9-s + (−0.604 − 1.04i)10-s + (−3.95 + 2.28i)11-s + (2.39 − 4.14i)12-s + (3.57 + 0.464i)13-s + (6.88 − 0.181i)14-s + 0.464i·15-s + (−4.66 + 8.07i)16-s + (−2.16 − 3.74i)17-s + ⋯
L(s)  = 1  + (1.59 + 0.920i)2-s + (−0.288 − 0.499i)3-s + (1.19 + 2.07i)4-s + (−0.179 − 0.103i)5-s − 1.06i·6-s + (0.852 − 0.522i)7-s + 2.56i·8-s + (−0.166 + 0.288i)9-s + (−0.191 − 0.331i)10-s + (−1.19 + 0.688i)11-s + (0.690 − 1.19i)12-s + (0.991 + 0.128i)13-s + (1.84 − 0.0483i)14-s + 0.119i·15-s + (−1.16 + 2.01i)16-s + (−0.525 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34022 + 1.29026i\)
\(L(\frac12)\) \(\approx\) \(2.34022 + 1.29026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 + 1.38i)T \)
13 \( 1 + (-3.57 - 0.464i)T \)
good2 \( 1 + (-2.25 - 1.30i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.401 + 0.232i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.95 - 2.28i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.65 + 1.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 + 3.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 + (8.13 - 4.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.35 - 3.66i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.95iT - 41T^{2} \)
43 \( 1 - 8.16T + 43T^{2} \)
47 \( 1 + (-1.67 - 0.966i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.31 - 5.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.00 - 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.41 + 4.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.89 - 2.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.37iT - 71T^{2} \)
73 \( 1 + (2.95 - 1.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.731iT - 83T^{2} \)
89 \( 1 + (2.72 + 1.57i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.96iT - 97T^{2} \)
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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

−12.46760674870312352115197168502, −11.37677456021054111877631778056, −10.73770264505275906367344448472, −8.647933981821291387287754581095, −7.60805712373726893053500200467, −7.08572046668554097467310513349, −5.89456894719566572055228097301, −4.92409866735597726096109183970, −4.13340549924226762649661506057, −2.43113505741050634599693966198, 1.95794151819401967779420111767, 3.40592628066501234519694669158, 4.32437169944598362921741590332, 5.60258792424065135983239693837, 5.88420375541199608665752314609, 7.79155022602732595126226567408, 9.132739673163142974263555578616, 10.61344235926629133503298804410, 11.05533855581991585928175153913, 11.54889421937691239839380672148

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