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Properties

Label 2-273-91.74-c1-0-11
Degree $2$
Conductor $273$
Sign $0.946 - 0.323i$
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  + 2.05·2-s + (0.5 + 0.866i)3-s + 2.22·4-s + (−0.274 − 0.475i)5-s + (1.02 + 1.77i)6-s + (2.59 + 0.527i)7-s + 0.456·8-s + (−0.499 + 0.866i)9-s + (−0.564 − 0.977i)10-s + (−2.34 − 4.06i)11-s + (1.11 + 1.92i)12-s + (−0.663 + 3.54i)13-s + (5.32 + 1.08i)14-s + (0.274 − 0.475i)15-s − 3.50·16-s − 0.603·17-s + ⋯
L(s)  = 1  + 1.45·2-s + (0.288 + 0.499i)3-s + 1.11·4-s + (−0.122 − 0.212i)5-s + (0.419 + 0.726i)6-s + (0.979 + 0.199i)7-s + 0.161·8-s + (−0.166 + 0.288i)9-s + (−0.178 − 0.309i)10-s + (−0.708 − 1.22i)11-s + (0.320 + 0.555i)12-s + (−0.184 + 0.982i)13-s + (1.42 + 0.289i)14-s + (0.0709 − 0.122i)15-s − 0.876·16-s − 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.946 - 0.323i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.946 - 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70635 + 0.450350i\)
\(L(\frac12)\) \(\approx\) \(2.70635 + 0.450350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 0.527i)T \)
13 \( 1 + (0.663 - 3.54i)T \)
good2 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 + (0.274 + 0.475i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.34 + 4.06i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.603T + 17T^{2} \)
19 \( 1 + (0.280 - 0.485i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.376T + 23T^{2} \)
29 \( 1 + (-2.09 + 3.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.577 - 0.999i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.80T + 37T^{2} \)
41 \( 1 + (3.96 - 6.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.747 + 1.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.09 + 1.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.52 + 7.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.53T + 59T^{2} \)
61 \( 1 + (3.71 - 6.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.79 - 8.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.88 - 5.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.24 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 - 9.18T + 89T^{2} \)
97 \( 1 + (-3.15 - 5.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.90969090053810999398394895537, −11.39450179686526896727301432861, −10.35862624388132104106164996173, −8.872398993563301408469863375665, −8.206855217722597339604626291692, −6.67322512150191429225763125394, −5.44814630057996683606345906098, −4.74811778516330195704552670470, −3.72177639620086344223807054618, −2.42021904308716418993873679538, 2.12676436934481645113374256720, 3.38671145260066128958807052597, 4.75966617864777879363003197982, 5.41724387305438855431616730182, 6.90354253933115546225911032161, 7.60154904970995619121038999201, 8.778486359152685546173271165635, 10.29537891577595015177998481402, 11.20699992775944011521677019642, 12.29269105365733184428775686212

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