Properties

Label 1-273-273.122-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.810 + 0.586i$
Analytic cond. $29.3379$
Root an. cond. $29.3379$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s i·8-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s i·8-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.810 + 0.586i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.810 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1340485799 + 0.4138773199i\)
\(L(\frac12)\) \(\approx\) \(0.1340485799 + 0.4138773199i\)
\(L(1)\) \(\approx\) \(0.6656374723 + 0.03373457555i\)
\(L(1)\) \(\approx\) \(0.6656374723 + 0.03373457555i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−25.20358814832729694427527405319, −24.43908271907825894099733949237, −23.63899481305059229358264483101, −22.52139656703184092477684499442, −21.061732997900560251894280959699, −20.70485776293821349239611127205, −19.42973606759423678235227207016, −18.45364824396628650337947295659, −17.841374424453177293392647417515, −16.65500708384865639237454845491, −16.28929276169516796503207601448, −15.0092048640902064204355423760, −14.03404914592097471181497895743, −13.08376393690033077689348336619, −11.77656209275399880022625584613, −10.472922373824191010681754313737, −9.887833978264046917564187793961, −8.731745952138809322089914200350, −8.01913027565739456557384551415, −6.676808116758288472811656938804, −5.73546058981700761667451648619, −4.84031567208757686782801284392, −2.748406822492671452935365553317, −1.549517037467580335716915398017, −0.16759993089686253962632650947, 1.66489947305575929006787953420, 2.51420121158221436728448396461, 3.79445106265535082726694413252, 5.490025849834652281878122445848, 6.68680620499783787896755251278, 7.68668085093262011624174907121, 8.7790208969938251873585655724, 9.9336290984125364451032992457, 10.42046444668915626242068125584, 11.47470253082485221992805164676, 12.72296084277740598729823716216, 13.430093891421890168666432280105, 14.83919983185452786600752117775, 15.7615613510529650254306811882, 17.09505429336104175100053780222, 17.56046503950878104566510764514, 18.55185597148864993104825801129, 19.24535741738779529658317158439, 20.41203901691424820609164108172, 21.2577481487249143223864897149, 21.83834528865094084354309944226, 23.00322730272905515129797994911, 24.216299984973977862366948515732, 25.37667958493939356769149896304, 25.989062120156396346238755753426

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