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Properties

Label 2-273-273.269-c1-0-11
Degree $2$
Conductor $273$
Sign $0.618 - 0.785i$
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  + 1.40i·2-s + (−0.386 − 1.68i)3-s + 0.0248·4-s + (1.48 + 2.56i)5-s + (2.37 − 0.543i)6-s + (0.0176 − 2.64i)7-s + 2.84i·8-s + (−2.70 + 1.30i)9-s + (−3.61 + 2.08i)10-s + (2.35 − 1.35i)11-s + (−0.00962 − 0.0420i)12-s + (3.51 + 0.787i)13-s + (3.71 + 0.0247i)14-s + (3.76 − 3.49i)15-s − 3.94·16-s − 0.466·17-s + ⋯
L(s)  = 1  + 0.993i·2-s + (−0.223 − 0.974i)3-s + 0.0124·4-s + (0.663 + 1.14i)5-s + (0.968 − 0.221i)6-s + (0.00666 − 0.999i)7-s + 1.00i·8-s + (−0.900 + 0.435i)9-s + (−1.14 + 0.659i)10-s + (0.710 − 0.410i)11-s + (−0.00277 − 0.0121i)12-s + (0.975 + 0.218i)13-s + (0.993 + 0.00661i)14-s + (0.972 − 0.903i)15-s − 0.987·16-s − 0.113·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32929 + 0.645421i\)
\(L(\frac12)\) \(\approx\) \(1.32929 + 0.645421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.386 + 1.68i)T \)
7 \( 1 + (-0.0176 + 2.64i)T \)
13 \( 1 + (-3.51 - 0.787i)T \)
good2 \( 1 - 1.40iT - 2T^{2} \)
5 \( 1 + (-1.48 - 2.56i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.35 + 1.35i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.466T + 17T^{2} \)
19 \( 1 + (-2.24 - 1.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.42iT - 23T^{2} \)
29 \( 1 + (7.22 + 4.16i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.80 - 3.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (-4.19 + 7.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.26 + 7.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.995 + 0.574i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + (-7.90 - 4.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.22 + 3.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.31 + 4.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.75 + 3.90i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.45 + 2.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 6.88T + 89T^{2} \)
97 \( 1 + (8.95 - 5.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.86733586457239177532684394493, −11.12659739722687766582898397244, −10.37194664576431354155280228338, −8.830099012512775937692187102062, −7.71290409141667386455781998130, −6.91473843274505206409378487150, −6.40703639845237178824847210430, −5.53961027890947002867078075889, −3.44083268419699906965490856763, −1.79200612288875407497191012113, 1.51062325577942966054041110477, 3.07537378861528904472190379119, 4.38311861659363898512617687934, 5.46842608156849947535824584588, 6.43023946578102975661711008603, 8.489329225006715271101517392947, 9.298304214512232392462998080680, 9.758482853816858939593020023507, 10.96299521261406527268413267408, 11.65872889459368106959784211336

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