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Properties

Label 2-273-91.73-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.264 - 0.964i$
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  + (−0.246 + 0.918i)2-s + (−0.866 − 0.5i)3-s + (0.949 + 0.548i)4-s + (0.797 + 0.213i)5-s + (0.672 − 0.672i)6-s + (−2.22 + 1.42i)7-s + (−2.08 + 2.08i)8-s + (0.499 + 0.866i)9-s + (−0.392 + 0.679i)10-s + (0.430 + 1.60i)11-s + (−0.548 − 0.949i)12-s + (3.57 + 0.481i)13-s + (−0.759 − 2.39i)14-s + (−0.583 − 0.583i)15-s + (−0.303 − 0.525i)16-s + (−1.31 + 2.28i)17-s + ⋯
L(s)  = 1  + (−0.174 + 0.649i)2-s + (−0.499 − 0.288i)3-s + (0.474 + 0.274i)4-s + (0.356 + 0.0955i)5-s + (0.274 − 0.274i)6-s + (−0.842 + 0.538i)7-s + (−0.735 + 0.735i)8-s + (0.166 + 0.288i)9-s + (−0.124 + 0.214i)10-s + (0.129 + 0.484i)11-s + (−0.158 − 0.274i)12-s + (0.991 + 0.133i)13-s + (−0.202 − 0.640i)14-s + (−0.150 − 0.150i)15-s + (−0.0758 − 0.131i)16-s + (−0.319 + 0.553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.264 - 0.964i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644151 + 0.844244i\)
\(L(\frac12)\) \(\approx\) \(0.644151 + 0.844244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.22 - 1.42i)T \)
13 \( 1 + (-3.57 - 0.481i)T \)
good2 \( 1 + (0.246 - 0.918i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.797 - 0.213i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.430 - 1.60i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.31 - 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.01 - 1.61i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.64 - 3.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.62T + 29T^{2} \)
31 \( 1 + (0.0712 + 0.265i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-11.4 - 3.07i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.98 + 6.98i)T - 41iT^{2} \)
43 \( 1 + 2.91iT - 43T^{2} \)
47 \( 1 + (-2.05 + 7.66i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.41 + 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.05 - 1.89i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.64 + 3.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.26 - 1.94i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.0 - 11.0i)T + 71iT^{2} \)
73 \( 1 + (2.81 - 0.753i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 + 10.9i)T - 83iT^{2} \)
89 \( 1 + (-0.689 + 2.57i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.73 - 4.73i)T - 97iT^{2} \)
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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

−12.02120772073795216915361875022, −11.50927979354886918739545464449, −10.20123091997312917212606770437, −9.275265431840209370474533368280, −8.101492957169389691889537480530, −7.17339136048125349178606499993, −6.07044628914130646810708341537, −5.77126101190741821928554014532, −3.70564920396549041221425301618, −2.12203903365417456492960096796, 0.941017298511840111158518989970, 2.85433769346762917397727308357, 4.05821303447567151722844320669, 5.83844881795091596502813290709, 6.33745951428476263466737372843, 7.63480180466602330828739790254, 9.370597954211438255477590102892, 9.729098660168505667541163275175, 10.85936350412849387996061441897, 11.33898380320768233400784549052

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