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Properties

Label 1-273-273.257-r0-0-0
Degree $1$
Conductor $273$
Sign $0.927 + 0.374i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  + 2-s + 4-s + (0.5 + 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 16-s + 17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 4-s + (0.5 + 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 16-s + 17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.412899711 + 0.4690856261i\)
\(L(\frac12)\) \(\approx\) \(2.412899711 + 0.4690856261i\)
\(L(1)\) \(\approx\) \(1.974956782 + 0.2282725771i\)
\(L(1)\) \(\approx\) \(1.974956782 + 0.2282725771i\)

Euler product

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

−25.53407967226194095416395172711, −24.59455251951113730397316124458, −23.7658019639427937019436261384, −23.08173449724473522509713769878, −21.887081669207775978809410430540, −21.23152117327349309479319379040, −20.35692267887874687288325552849, −19.71242879014442997361933930165, −18.22652803898206997238574077969, −17.12790833395704030711434565822, −16.297200575215240674580167741756, −15.37367641139748903016522708838, −14.375769688522254731481911943333, −13.39933699438816453800336559494, −12.63284671413707962715637468720, −11.91315308913042173846699237760, −10.55836705851273950980576023523, −9.64496542943816392607517558898, −8.24067558884500497527961933480, −7.14514325035651326371196587205, −5.9101454143574864846547981372, −5.04458941050903129935097575068, −4.157045135444236194831868344145, −2.66003053655980546698228287846, −1.533321521405612395296018474228, 1.82687399004739372206766210842, 2.98263686421173500008818484705, 3.87261286085133974853536228782, 5.465735478564525367735596987498, 6.06475335094636524115340795234, 7.19791909295807911609061583850, 8.260687722126872890785474954269, 10.06448271404097554068994536792, 10.656030688468376769397678486621, 11.76215661410033397215342977996, 12.727606914419680789903893179673, 13.96298992612011698682471287402, 14.23261084466025287583925168551, 15.43197804425441438606480588419, 16.299977478612533703981020188369, 17.36654494330226881440316003551, 18.65098519683341697309050778088, 19.31932557706000505887934992097, 20.705186827523612767201567517120, 21.36006173180843685050487565864, 22.11539826420894403376302801031, 23.02661883102827812065755947957, 23.75233569978345764567661871490, 24.82317665637523888485973243854, 25.63037115409518782711705477057

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