Properties

Label 1-273-273.116-r1-0-0
Degree $1$
Conductor $273$
Sign $0.0633 + 0.997i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8019154646 + 0.7526388737i\)
\(L(\frac12)\) \(\approx\) \(0.8019154646 + 0.7526388737i\)
\(L(1)\) \(\approx\) \(0.6814184290 + 0.3377976874i\)
\(L(1)\) \(\approx\) \(0.6814184290 + 0.3377976874i\)

Euler product

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

−25.41715237339892208933058113949, −24.44912007457488554864646876479, −23.1687044086700281844644886926, −22.62968625699880380375046297065, −21.186081062472479637170655142050, −20.64945766151525735418880398782, −19.92561061412799268081976328408, −18.91910076213101557798604870617, −18.08551295834042195739760901235, −17.026217770136312559438389314397, −16.28440179505218972122352776726, −15.188589708672408012876755344281, −13.70541307092469434674851843665, −12.80801393261716930944585131815, −12.0245187350547464995592906045, −11.22252976086444524569541076366, −9.86110877193662523212032638074, −9.29071305770428924000668566282, −7.97526298950340937004711325195, −7.427568009122795301003527194642, −5.37644897500602522835270926160, −4.39890405635421163466579721281, −3.27050035107309866609716226308, −1.85031907332267840417706825090, −0.62177308835489028184500136325, 0.79729764721385682354658922599, 2.70455487940345036406053612416, 4.07508954892929689918102707306, 5.46891175418724339926530959499, 6.43602279762418527869326707669, 7.44324940799011500071960928689, 8.24206900209650725077477534766, 9.34190399427151491032761727255, 10.61709247133116891242508798186, 11.11194174221895400831692185051, 12.73213682479877617894361968830, 13.9423382791859893893823307868, 14.683197763982429650749009440793, 15.60480069210832195454740555372, 16.339196894279266246891752695575, 17.44109928544863121237900043710, 18.38356288667757282788986279808, 19.06139199183642708509433218845, 19.81526169688139082111008529519, 21.30342525291083307093218165867, 22.34539957678255639331100980732, 23.1630654533417346660595280669, 23.978530524587501714565525255055, 24.74814576781331566387779150295, 26.078080003221768444415913901486

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