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Properties

Label 2-273-91.16-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.781 + 0.624i$
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  − 2.55·2-s + (−0.5 + 0.866i)3-s + 4.50·4-s + (−1.40 + 2.44i)5-s + (1.27 − 2.20i)6-s + (−2.27 + 1.35i)7-s − 6.39·8-s + (−0.499 − 0.866i)9-s + (3.59 − 6.22i)10-s + (−0.265 + 0.460i)11-s + (−2.25 + 3.90i)12-s + (2.71 + 2.36i)13-s + (5.80 − 3.44i)14-s + (−1.40 − 2.44i)15-s + 7.29·16-s + 0.405·17-s + ⋯
L(s)  = 1  − 1.80·2-s + (−0.288 + 0.499i)3-s + 2.25·4-s + (−0.630 + 1.09i)5-s + (0.520 − 0.901i)6-s + (−0.859 + 0.511i)7-s − 2.26·8-s + (−0.166 − 0.288i)9-s + (1.13 − 1.96i)10-s + (−0.0800 + 0.138i)11-s + (−0.650 + 1.12i)12-s + (0.753 + 0.657i)13-s + (1.55 − 0.921i)14-s + (−0.363 − 0.630i)15-s + 1.82·16-s + 0.0984·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.781 + 0.624i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0344417 - 0.0982295i\)
\(L(\frac12)\) \(\approx\) \(0.0344417 - 0.0982295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 - 1.35i)T \)
13 \( 1 + (-2.71 - 2.36i)T \)
good2 \( 1 + 2.55T + 2T^{2} \)
5 \( 1 + (1.40 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.265 - 0.460i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.405T + 17T^{2} \)
19 \( 1 + (1.83 + 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 + (4.44 + 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.44 + 7.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 + (-5.44 - 9.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.52 - 7.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.45 + 7.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.51 + 6.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0728T + 59T^{2} \)
61 \( 1 + (-4.87 - 8.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.98 - 6.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.535 - 0.928i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.729 - 1.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.53 + 6.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.449T + 83T^{2} \)
89 \( 1 + 0.922T + 89T^{2} \)
97 \( 1 + (0.0841 - 0.145i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.63274172082270315068675120304, −11.38676667893908748421328162570, −10.39003495978796119400801358701, −9.650348242863530910701591597538, −8.862804380076824474231420793463, −7.76510507090328298029079610244, −6.78354328815089655690744089741, −6.06924297885560756441831983187, −3.75600574211291292416181687288, −2.42077509127140124896564883790, 0.14802010370159687387793001357, 1.46641617011552300022191161377, 3.55194484361793376114006389274, 5.59109070973677541465083487670, 6.81287155447255346574181046078, 7.65515944759342382431074010481, 8.543139638756379409579997881743, 9.140740842314364423633190353354, 10.45387263516694212418019725086, 10.88775458082345433869213666486

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