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Properties

Label 2-273-273.68-c1-0-28
Degree $2$
Conductor $273$
Sign $0.331 + 0.943i$
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  + 0.546i·2-s + (−0.761 − 1.55i)3-s + 1.70·4-s + (1.25 − 2.17i)5-s + (0.849 − 0.416i)6-s + (−0.612 − 2.57i)7-s + 2.02i·8-s + (−1.83 + 2.37i)9-s + (1.18 + 0.686i)10-s + (−5.22 − 3.01i)11-s + (−1.29 − 2.64i)12-s + (1.89 + 3.06i)13-s + (1.40 − 0.334i)14-s + (−4.34 − 0.296i)15-s + 2.29·16-s − 0.647·17-s + ⋯
L(s)  = 1  + 0.386i·2-s + (−0.439 − 0.898i)3-s + 0.850·4-s + (0.561 − 0.973i)5-s + (0.346 − 0.169i)6-s + (−0.231 − 0.972i)7-s + 0.714i·8-s + (−0.612 + 0.790i)9-s + (0.375 + 0.217i)10-s + (−1.57 − 0.908i)11-s + (−0.374 − 0.764i)12-s + (0.526 + 0.850i)13-s + (0.375 − 0.0894i)14-s + (−1.12 − 0.0765i)15-s + 0.574·16-s − 0.156·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08359 - 0.768086i\)
\(L(\frac12)\) \(\approx\) \(1.08359 - 0.768086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.761 + 1.55i)T \)
7 \( 1 + (0.612 + 2.57i)T \)
13 \( 1 + (-1.89 - 3.06i)T \)
good2 \( 1 - 0.546iT - 2T^{2} \)
5 \( 1 + (-1.25 + 2.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.22 + 3.01i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.647T + 17T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.03iT - 23T^{2} \)
29 \( 1 + (-4.46 + 2.57i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.25 + 5.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 + (-4.48 - 7.77i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.09 - 7.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.09 + 3.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0406 + 0.0234i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.66T + 59T^{2} \)
61 \( 1 + (3.11 - 1.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.761 - 1.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.4 - 6.62i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.76 + 2.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.14 - 3.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 1.01T + 89T^{2} \)
97 \( 1 + (8.58 + 4.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.63720958009659492976452669955, −10.99409543618863394292364632498, −9.965347588220153779680478465857, −8.403156248338923580204983305612, −7.78476932726786256105167441044, −6.65823116092625079777408715417, −5.89606525956854030811978703277, −4.87076424719078018039018523471, −2.70470614150859129782848344581, −1.13221007894961397443562662216, 2.49095663222139737807598242096, 3.17862935720565567065456974106, 5.11473035873486468141401519991, 5.99559221975512183480252527857, 6.92490329518123810391948277662, 8.321410699480761795509515624810, 9.757142587515229333789836830530, 10.47026433678583017906240785369, 10.73802193476037886903900774688, 12.05024717876947192707935955791

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