Properties

Label 2-273-273.80-c0-0-0
Degree $2$
Conductor $273$
Sign $-0.100 - 0.994i$
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  + i·3-s + (−0.866 + 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (−0.866 − 0.5i)21-s + (−0.866 − 0.5i)25-s i·27-s − 0.999i·28-s + (1.36 − 0.366i)31-s + (0.866 − 0.5i)36-s + (0.5 − 0.133i)37-s + ⋯
L(s)  = 1  + i·3-s + (−0.866 + 0.5i)4-s + (−0.5 + 0.866i)7-s − 9-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)13-s + (0.499 − 0.866i)16-s + (1.36 + 1.36i)19-s + (−0.866 − 0.5i)21-s + (−0.866 − 0.5i)25-s i·27-s − 0.999i·28-s + (1.36 − 0.366i)31-s + (0.866 − 0.5i)36-s + (0.5 − 0.133i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(0.136244\)
Root analytic conductor: \(0.369113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :0),\ -0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6086649003\)
\(L(\frac12)\) \(\approx\) \(0.6086649003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−12.22157640284510323689059769178, −11.60352512859213456000844825590, −10.12765993759472145699502311956, −9.650958419204735814008440258436, −8.609382485427544086735652332058, −7.980561807734962318510237680934, −6.04903113434626166223002293018, −5.25547027501461647727713418139, −3.93353428475640900203245746300, −3.06952599328811000904381042252, 1.16752905340420434536210547571, 3.29030769500172323723293504840, 4.71738145121298426168457696069, 6.00608123910532792126270461327, 6.90373742155447837391133364102, 7.972053878432931158513995716845, 9.033668792989128623306379253608, 9.861941906027809375403217963769, 11.04152999738188505229706027754, 11.89451732075340779749753212166

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