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Properties

Label 1-273-273.263-r1-0-0
Degree $1$
Conductor $273$
Sign $0.949 - 0.313i$
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 + 10-s − 11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.5 + 0.866i)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 + (−0.5 − 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 + 10-s − 11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.5 + 0.866i)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 + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.017587176 - 0.3243166239i\)
\(L(\frac12)\) \(\approx\) \(2.017587176 - 0.3243166239i\)
\(L(1)\) \(\approx\) \(1.289774281 + 0.2679502958i\)
\(L(1)\) \(\approx\) \(1.289774281 + 0.2679502958i\)

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 \( 1 - T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
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 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−25.64364515394006450899777170470, −24.41541161924455810313027947335, −23.43525006106548931682881803609, −22.704903200672051094308952281, −21.77756437936940916389498294542, −21.15463802900406081593830575123, −20.190512267213735372947239156518, −19.103779332235864282844961119460, −18.36449001937715853108044869698, −17.66410065152273770905571800200, −16.12705106538839629490785729814, −14.93425869116529289396353828027, −14.28272266875898266471114534394, −13.29038961515711574550061887855, −12.472401458359419979552557803531, −11.24306025308677378891067830749, −10.45181904830047813945978789096, −9.78855288854244502346291959786, −8.4247576988669418364249818976, −6.96424022102329053271278623209, −5.82104844052629442830232108026, −4.90437164284411319078473114127, −3.38237082221654179663562589912, −2.640706507031475806735522923, −1.31371042208338518080498781270, 0.561228181057559454870140900195, 2.48820604007981776898641099339, 3.88097711710668738109649131729, 5.281681687982364240516852970265, 5.51985619421060801545667895513, 7.114710362548041414570212346734, 7.966269221971018760200565741175, 9.062362045714410660498606208606, 9.926622968982147873051304763057, 11.60652233759857883988648930689, 12.57427859351963709986139824941, 13.468019067814047927932652615627, 14.057942876379270550130317850326, 15.469197034638504781214324624459, 16.055461777544200165468713303173, 17.01171296610047562350797461250, 17.819871527794500529168436298850, 18.74706596428036786517672955160, 20.35794981290993775582765512766, 20.99679966717458726441263370464, 21.83772777423640125089398767611, 22.9469883928535006029096332139, 23.69703657099995594957036314169, 24.60308726351423793534755861938, 25.22052619566339048461617187344

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