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Properties

Label 2-273-13.10-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.828 + 0.560i$
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  + (−1.98 + 1.14i)2-s + (0.5 + 0.866i)3-s + (1.62 − 2.82i)4-s − 0.692i·5-s + (−1.98 − 1.14i)6-s + (−0.866 − 0.5i)7-s + 2.88i·8-s + (−0.499 + 0.866i)9-s + (0.793 + 1.37i)10-s + (−5.20 + 3.00i)11-s + 3.25·12-s + (−3.26 − 1.53i)13-s + 2.29·14-s + (0.599 − 0.346i)15-s + (−0.0518 − 0.0897i)16-s + (−3.99 + 6.92i)17-s + ⋯
L(s)  = 1  + (−1.40 + 0.810i)2-s + (0.288 + 0.499i)3-s + (0.814 − 1.41i)4-s − 0.309i·5-s + (−0.810 − 0.468i)6-s + (−0.327 − 0.188i)7-s + 1.02i·8-s + (−0.166 + 0.288i)9-s + (0.251 + 0.434i)10-s + (−1.57 + 0.906i)11-s + 0.940·12-s + (−0.905 − 0.425i)13-s + 0.612·14-s + (0.154 − 0.0893i)15-s + (−0.0129 − 0.0224i)16-s + (−0.969 + 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.828 + 0.560i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.828 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0513810 - 0.167529i\)
\(L(\frac12)\) \(\approx\) \(0.0513810 - 0.167529i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (3.26 + 1.53i)T \)
good2 \( 1 + (1.98 - 1.14i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 0.692iT - 5T^{2} \)
11 \( 1 + (5.20 - 3.00i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.99 - 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.60 + 1.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.326 - 0.565i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.47iT - 31T^{2} \)
37 \( 1 + (3.49 - 2.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.95 + 4.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.06 - 7.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.664iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (1.76 + 1.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.693 - 1.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.39 - 3.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.867 - 0.501i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 - 6.08T + 79T^{2} \)
83 \( 1 + 0.377iT - 83T^{2} \)
89 \( 1 + (5.74 - 3.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.299 + 0.173i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.65757995486342217281205120164, −10.83280521522460235052967955549, −10.29308033249970423685920029658, −9.553014516835111658735037650448, −8.548719692923648906244098016636, −7.84869490958410277120764795235, −6.93187301305041075292715235025, −5.65122205586224763578299672651, −4.37224180843267643646543315576, −2.31456796659939066909557379335, 0.18461574104759830359332073419, 2.30396450128063759905625856705, 3.01517120750339124176614306717, 5.22812409875030146356076652968, 6.91515129384007909450443463139, 7.63167489452659861080303939565, 8.715806493135511125563697307317, 9.313848781249830995452429157999, 10.47163436934876298729196567620, 11.04812638403989804737708827813

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