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Properties

Label 2-273-13.10-c1-0-7
Degree $2$
Conductor $273$
Sign $0.631 - 0.775i$
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  + (0.924 − 0.533i)2-s + (0.5 + 0.866i)3-s + (−0.430 + 0.745i)4-s + 0.994i·5-s + (0.924 + 0.533i)6-s + (0.866 + 0.5i)7-s + 3.05i·8-s + (−0.499 + 0.866i)9-s + (0.530 + 0.919i)10-s + (−0.215 + 0.124i)11-s − 0.860·12-s + (−3.56 − 0.529i)13-s + 1.06·14-s + (−0.860 + 0.497i)15-s + (0.769 + 1.33i)16-s + (3.10 − 5.37i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.377i)2-s + (0.288 + 0.499i)3-s + (−0.215 + 0.372i)4-s + 0.444i·5-s + (0.377 + 0.217i)6-s + (0.327 + 0.188i)7-s + 1.07i·8-s + (−0.166 + 0.288i)9-s + (0.167 + 0.290i)10-s + (−0.0650 + 0.0375i)11-s − 0.248·12-s + (−0.989 − 0.146i)13-s + 0.285·14-s + (−0.222 + 0.128i)15-s + (0.192 + 0.333i)16-s + (0.752 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.631 - 0.775i$
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: \((2,\ 273,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61031 + 0.764887i\)
\(L(\frac12)\) \(\approx\) \(1.61031 + 0.764887i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.56 + 0.529i)T \)
good2 \( 1 + (-0.924 + 0.533i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 0.994iT - 5T^{2} \)
11 \( 1 + (0.215 - 0.124i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.10 + 5.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.28 - 4.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.907 + 1.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.34 + 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.53iT - 31T^{2} \)
37 \( 1 + (3.66 - 2.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.57 + 3.21i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.71 - 9.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + 0.601T + 53T^{2} \)
59 \( 1 + (7.71 + 4.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.87 - 4.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.73 + 3.31i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.4 - 6.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.70iT - 73T^{2} \)
79 \( 1 + 5.63T + 79T^{2} \)
83 \( 1 + 16.5iT - 83T^{2} \)
89 \( 1 + (8.47 - 4.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.10 - 1.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.90415115896720172561142641539, −11.44525688243233719588643798882, −10.09636955394070426595318397427, −9.387076582688803430612710291332, −8.057711228987438188039566238852, −7.37860916359488873979378671446, −5.53575342336017049397703089183, −4.76156505754206309446520176625, −3.45507190714334913256543121503, −2.55837453557275015142563718932, 1.30027106143439142890700495617, 3.33944192466494705197134553226, 4.78344183299259218750180192738, 5.51333941367982064165923186739, 6.81889175336447890753830842084, 7.65194647095589810806352531479, 8.893611133017276410792124011485, 9.761033520791199629884848530722, 10.83688133239995747507969965189, 12.24456115666942736931115992896

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