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Properties

Label 1-273-273.227-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.574 + 0.818i$
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  + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + 16-s − 17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + 16-s − 17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9618082779 + 1.849580882i\)
\(L(\frac12)\) \(\approx\) \(0.9618082779 + 1.849580882i\)
\(L(1)\) \(\approx\) \(0.9769231505 + 0.7475497203i\)
\(L(1)\) \(\approx\) \(0.9769231505 + 0.7475497203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + iT \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 - T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + iT \)
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 \( 1 + iT \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + iT \)
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

−25.00168830679370849116676943091, −24.40958237698069602909915920819, −23.04687767019541994225619954318, −22.23245035553352724903302252819, −21.42228879039483971669770092816, −20.69133552992103421343906987789, −19.77202796838880083402108148915, −18.93622641854529518472249946455, −17.757788030816340760285442383012, −17.24723843701562311503611894260, −16.08156306001485700335127959371, −14.51500387782448398291173394876, −13.747042890315406793294592115881, −12.96816847485290132770962477960, −11.93848193013859095028161303690, −11.033588376632089507607769115631, −9.89469713384652456360385576396, −9.15266102706813748496456601091, −8.296618690941979347877573335939, −6.53201107774531945657938761859, −5.37536028532688737182071446059, −4.35161612667161071809721586263, −3.06428397897972884672069133214, −1.814575637280453708827893756007, −0.75526326491106248182795358507, 1.24438485838543435714268691626, 2.93284417106902083148011714410, 4.4121608611177867118750258166, 5.43635764688596663570117608089, 6.63823876292892967158186915524, 7.07825084278127301480639339580, 8.64053527169262525603677742790, 9.42337421765117341904056890947, 10.34919297577594305437214158671, 11.734079812393490710716337169382, 13.1325266377536635389642412021, 13.77749536306150390700771842870, 14.7526105130523870359306517643, 15.454886169084791293359346059932, 16.71479753687630840808409233860, 17.4783816475389790794066350554, 18.120131091914504575445009071904, 19.142191356315058838262829956378, 20.327279851491940889121935012491, 21.67659613650399828423056224987, 22.281116344184226518646242117823, 23.02322591255795779590125352269, 24.2545398212982778360627894084, 24.947212894332636009972307035151, 25.68154060752041105998987147238

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