Properties

Label 2-273-273.74-c0-0-1
Degree $2$
Conductor $273$
Sign $0.656 + 0.754i$
Analytic cond. $0.136244$
Root an. cond. $0.369113$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (−0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7496587858\)
\(L(\frac12)\) \(\approx\) \(0.7496587858\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.96769113514174420928488581328, −11.16007466888114300179655769407, −10.44743648707332786626995305341, −9.180034034704280224597572975722, −7.62016775293943546341621076203, −7.07462944416210572592329253040, −6.34453796924869618127085803525, −5.03911227863593773966053577509, −3.23171219855847535167587717600, −1.69426267454846610406584398035, 2.54094456523031923158939771246, 3.75847791594342867408926511147, 5.50466788895028055692504694322, 5.96142704152549843878073701497, 7.26895720981183049590510254049, 8.494182551239677130502888253345, 9.759334153033061636597031405850, 10.29239079010745481225468279534, 11.38895877606578105374239008950, 12.06213236697114044589849094033

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