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Properties

Label 1-273-273.149-r0-0-0
Degree $1$
Conductor $273$
Sign $0.0247 - 0.999i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s − i·8-s − 10-s − i·11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)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 − 10-s − i·11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0247 - 0.999i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 0.0247 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6412391743 - 0.6255545486i\)
\(L(\frac12)\) \(\approx\) \(0.6412391743 - 0.6255545486i\)
\(L(1)\) \(\approx\) \(0.7518807877 - 0.3319741547i\)
\(L(1)\) \(\approx\) \(0.7518807877 - 0.3319741547i\)

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 \( 1 - iT \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
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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

−26.2054530882528338242495891576, −25.04392304193149617658981890637, −24.5633870614463793794801728485, −23.23032887307601961741311500005, −22.50369203057965364795405065753, −21.20644441448016229179403608794, −20.389074403353241366292983411016, −19.29743630366348208807962238012, −18.42497460078866133783772458920, −17.597373372625651610311060786957, −17.01983052546217869627115099926, −15.80592489779793054014868234882, −14.84182346014391746633537823412, −14.17172671290224603588304507721, −12.88982368008412769049354453080, −11.585616775356365839649071617691, −10.25034485634869751986247931728, −9.99769089859447145146226057831, −8.72537687790001456925238430166, −7.68854114443200683926395565427, −6.542180240885990329969484637324, −5.91167567238745904882484295262, −4.483579992558503369294738296526, −2.53788282043862622377822094179, −1.58667492747111487682042701626, 0.8426368643733057639180196844, 2.17175964841668402621303866393, 3.279148620049687250131417373778, 4.86429710083287047075340656597, 6.16769855139541600452265780674, 7.28219463581934521712466064708, 8.64427836349167550308556838521, 9.1556483092877164248949731255, 10.238564159400838489191693165598, 11.17399145332327394876488148295, 12.1668945497147101297279748370, 13.286882741966196810354302068190, 13.98106235447122004172062877997, 15.76181476136904258288732381348, 16.34628835555669305469047837537, 17.53664112796503422673479945504, 17.9162102581327885754722359966, 19.156329926023574231824645184070, 19.92217127321722234448128192907, 20.939350501657659532557457605329, 21.57409230061752530723650919284, 22.393036019103416227924983462925, 24.04823478870467271391399419934, 24.71608863945572131931737520097, 25.65862767472783503678589846220

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