Properties

Label 1-273-273.89-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: $\chi_{273} (89, \cdot )$
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

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

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