↑

Properties

Label 1-273-273.269-r0-0-0
Degree $1$
Conductor $273$
Sign $0.617 + 0.786i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6500303395 + 0.3163032409i\)
\(L(\frac12)\) \(\approx\) \(0.6500303395 + 0.3163032409i\)
\(L(1)\) \(\approx\) \(0.6754022739 + 0.1330154178i\)
\(L(1)\) \(\approx\) \(0.6754022739 + 0.1330154178i\)

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 \( 1 + (0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.58609909993732137533743571321, −24.83212760030429502955077793989, −23.99725383782668594032574956129, −23.0873677142701258318582612162, −21.71768691557134121143930159838, −20.63847234751932228802327513042, −20.02929121078341666259881661041, −19.27710867214741548102473816250, −18.17312651210501503245523600088, −17.25784844231697529985457164183, −16.509604309692710153437167274206, −15.62081378869510534527895343022, −14.76241965552914672046446127850, −13.24060774605023324266343928416, −11.99047132782761404454381944499, −11.64024688683709062515210353683, −10.055499317098261976952621304236, −9.43736607486645934844156808741, −8.28378077534950849630145973744, −7.57513684145079964453258412867, −6.38861099676114486986075489971, −5.03555866014568899521030543908, −3.70474078956374546308092440310, −2.12714148490022862041544786744, −0.80945037253501549033619631848, 1.22925007464441565112802509741, 2.834354151451070647836342288652, 3.71313576576259494206661840746, 5.74667893088172827988264290804, 6.66942617856956132177655683361, 7.72890142773895497539248577852, 8.47936797446181387517350021794, 9.8047425917441314121979683941, 10.561151568337196593742091879043, 11.577038412395513923302295991935, 12.24062565822528085889347645503, 14.06858814368496433025147974312, 14.741924830274488820051938580628, 15.997464659616927481097462374732, 16.5053957474659716461656579158, 17.79401226901632380320996459294, 18.5501356665792940863914775530, 19.28055525442354391393175429606, 20.04783773094088952490722648002, 21.22025190177327150515382876850, 22.0975285666243924751156403824, 23.25704707107697598250055497269, 24.15890773551578027172670244048, 25.19653810767860552984421901672, 25.94205999484284249930843205060

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