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Properties

Label 2-273-91.23-c1-0-5
Degree $2$
Conductor $273$
Sign $0.985 + 0.168i$
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  − 0.976i·2-s + (−0.5 + 0.866i)3-s + 1.04·4-s + (0.233 + 0.134i)5-s + (0.845 + 0.488i)6-s + (1.06 + 2.42i)7-s − 2.97i·8-s + (−0.499 − 0.866i)9-s + (0.131 − 0.228i)10-s + (0.741 + 0.428i)11-s + (−0.523 + 0.906i)12-s + (1.45 + 3.29i)13-s + (2.36 − 1.03i)14-s + (−0.233 + 0.134i)15-s − 0.812·16-s + 2.17·17-s + ⋯
L(s)  = 1  − 0.690i·2-s + (−0.288 + 0.499i)3-s + 0.523·4-s + (0.104 + 0.0603i)5-s + (0.345 + 0.199i)6-s + (0.401 + 0.915i)7-s − 1.05i·8-s + (−0.166 − 0.288i)9-s + (0.0416 − 0.0721i)10-s + (0.223 + 0.129i)11-s + (−0.151 + 0.261i)12-s + (0.402 + 0.915i)13-s + (0.632 − 0.277i)14-s + (−0.0603 + 0.0348i)15-s − 0.203·16-s + 0.527·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.985 + 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45007 - 0.122860i\)
\(L(\frac12)\) \(\approx\) \(1.45007 - 0.122860i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 - 2.42i)T \)
13 \( 1 + (-1.45 - 3.29i)T \)
good2 \( 1 + 0.976iT - 2T^{2} \)
5 \( 1 + (-0.233 - 0.134i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.741 - 0.428i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 + (-1.73 + 0.999i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.92T + 23T^{2} \)
29 \( 1 + (2.97 + 5.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0946 - 0.0546i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.49iT - 37T^{2} \)
41 \( 1 + (7.03 - 4.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.78 - 3.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.592 + 0.342i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.21 + 7.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.00iT - 59T^{2} \)
61 \( 1 + (5.48 + 9.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.83 - 2.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.5 + 7.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.56 - 2.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.782 - 1.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.98iT - 83T^{2} \)
89 \( 1 - 2.71iT - 89T^{2} \)
97 \( 1 + (8.93 + 5.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.65145664493312131542084258510, −11.22726790104294753363436161885, −10.02781647609509927642030643960, −9.369182520251647211432565704317, −8.159585580877323524695292733693, −6.74841041391879135418128188648, −5.83509155914530279044346898043, −4.49334021248269752542400633539, −3.15230279640586358210394432517, −1.78553276445211003005052734010, 1.47933175597452228431396741243, 3.37109795587309443833146615131, 5.16562778206638569854218561208, 5.98403562264359378341468715246, 7.19058677697324509664061184182, 7.64513786405933499570168238054, 8.728391913218481221191700065907, 10.28131957639628413823762362418, 11.01949408003321174562199077720, 11.83995746572173307155104560257

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