Properties

Label 2-273-273.230-c1-0-28
Degree $2$
Conductor $273$
Sign $-0.172 + 0.985i$
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  + (1.29 + 0.746i)2-s + (−1.42 − 0.981i)3-s + (0.113 + 0.196i)4-s − 2.73·5-s + (−1.11 − 2.33i)6-s + (1.84 − 1.89i)7-s − 2.64i·8-s + (1.07 + 2.80i)9-s + (−3.53 − 2.03i)10-s + (−4.80 − 2.77i)11-s + (0.0310 − 0.391i)12-s + (−2.92 − 2.11i)13-s + (3.79 − 1.07i)14-s + (3.89 + 2.68i)15-s + (2.20 − 3.81i)16-s + (1.56 + 2.71i)17-s + ⋯
L(s)  = 1  + (0.913 + 0.527i)2-s + (−0.823 − 0.566i)3-s + (0.0566 + 0.0981i)4-s − 1.22·5-s + (−0.453 − 0.952i)6-s + (0.696 − 0.717i)7-s − 0.935i·8-s + (0.357 + 0.933i)9-s + (−1.11 − 0.644i)10-s + (−1.44 − 0.835i)11-s + (0.00895 − 0.112i)12-s + (−0.810 − 0.585i)13-s + (1.01 − 0.288i)14-s + (1.00 + 0.692i)15-s + (0.550 − 0.953i)16-s + (0.380 + 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.172 + 0.985i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613149 - 0.729567i\)
\(L(\frac12)\) \(\approx\) \(0.613149 - 0.729567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 + 0.981i)T \)
7 \( 1 + (-1.84 + 1.89i)T \)
13 \( 1 + (2.92 + 2.11i)T \)
good2 \( 1 + (-1.29 - 0.746i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.73T + 5T^{2} \)
11 \( 1 + (4.80 + 2.77i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.56 - 2.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.83 + 2.79i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.778 + 0.449i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.87 - 3.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.36iT - 31T^{2} \)
37 \( 1 + (1.46 - 2.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.12 - 1.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.910 - 1.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.03T + 47T^{2} \)
53 \( 1 + 8.97iT - 53T^{2} \)
59 \( 1 + (1.59 + 2.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.62 + 0.938i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.70 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.21 + 1.85i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 6.48T + 83T^{2} \)
89 \( 1 + (-5.08 + 8.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.68 - 4.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.84235138812164097679382469966, −10.86768371495696130633315737607, −10.19070960284177072713260733177, −8.032350409676954314857170303909, −7.66322164151504666369243075545, −6.62019020915500083634288157551, −5.22145990673146343455045130496, −4.85779884707887861253242720781, −3.36293916310308251977294378687, −0.61022343194098507287303774347, 2.62543798803572077662092344297, 4.03619484101461910376372978493, 4.89266734064653189682776763824, 5.47960879266787357484548380431, 7.38453256605107325691901543630, 8.120561166964206813123136617376, 9.573463294772211814252720253127, 10.66015799503455959962677590728, 11.70141032347461347499814178054, 11.97047186064617476913039336796

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