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Properties

Label 2-273-13.9-c1-0-11
Degree $2$
Conductor $273$
Sign $-1$
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  + (1.18 − 2.05i)2-s + (0.5 − 0.866i)3-s + (−1.82 − 3.16i)4-s − 4.02·5-s + (−1.18 − 2.05i)6-s + (0.5 + 0.866i)7-s − 3.92·8-s + (−0.499 − 0.866i)9-s + (−4.78 + 8.29i)10-s + (1.63 − 2.83i)11-s − 3.65·12-s + (0.910 − 3.48i)13-s + 2.37·14-s + (−2.01 + 3.48i)15-s + (−1.01 + 1.75i)16-s + (0.188 + 0.326i)17-s + ⋯
L(s)  = 1  + (0.840 − 1.45i)2-s + (0.288 − 0.499i)3-s + (−0.912 − 1.58i)4-s − 1.80·5-s + (−0.485 − 0.840i)6-s + (0.188 + 0.327i)7-s − 1.38·8-s + (−0.166 − 0.288i)9-s + (−1.51 + 2.62i)10-s + (0.493 − 0.854i)11-s − 1.05·12-s + (0.252 − 0.967i)13-s + 0.635·14-s + (−0.520 + 0.900i)15-s + (−0.253 + 0.439i)16-s + (0.0457 + 0.0792i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52306i\)
\(L(\frac12)\) \(\approx\) \(1.52306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.910 + 3.48i)T \)
good2 \( 1 + (-1.18 + 2.05i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 4.02T + 5T^{2} \)
11 \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.188 - 0.326i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.77 - 3.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.948 - 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.33 + 7.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.33T + 31T^{2} \)
37 \( 1 + (2.01 - 3.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.32 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 7.75T + 53T^{2} \)
59 \( 1 + (-1.05 - 1.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.87 + 6.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.79 - 4.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.99 + 3.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.50T + 73T^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + (3.17 - 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 + 4.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.76544702288331593818059683636, −11.00470887568194829066776455476, −9.872929748328341556755349805056, −8.357545273876155913220685641963, −7.87757728860861122416237447691, −6.18865687689488090115080754141, −4.74678904748333587912981009639, −3.64462008834228693150872169353, −2.96596235019041213893525120197, −0.970504142698149759165400378650, 3.52235045553583954417208475145, 4.32797491109275325481141649353, 4.93980089316024543068307165808, 6.78588432394752831969086912991, 7.23151300441027167955305607904, 8.260934037346162988440165959347, 8.984997697948884105747703920331, 10.60700279128365456105543936501, 11.77110940733718819290160107377, 12.37819790407729118517491767116

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