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Properties

Label 1-1323-1323.877-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.916 - 0.400i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.797 + 0.603i)2-s + (0.270 − 0.962i)4-s + (−0.0249 − 0.999i)5-s + (0.365 + 0.930i)8-s + (0.623 + 0.781i)10-s + (−0.797 + 0.603i)11-s + (0.542 − 0.840i)13-s + (−0.853 − 0.521i)16-s + (−0.900 + 0.433i)17-s + 19-s + (−0.969 − 0.246i)20-s + (0.270 − 0.962i)22-s + (0.698 + 0.715i)23-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)2-s + (0.270 − 0.962i)4-s + (−0.0249 − 0.999i)5-s + (0.365 + 0.930i)8-s + (0.623 + 0.781i)10-s + (−0.797 + 0.603i)11-s + (0.542 − 0.840i)13-s + (−0.853 − 0.521i)16-s + (−0.900 + 0.433i)17-s + 19-s + (−0.969 − 0.246i)20-s + (0.270 − 0.962i)22-s + (0.698 + 0.715i)23-s + (−0.998 + 0.0498i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9204071512 - 0.1923286920i\)
\(L(\frac12)\) \(\approx\) \(0.9204071512 - 0.1923286920i\)
\(L(1)\) \(\approx\) \(0.7430658896 + 0.01166850913i\)
\(L(1)\) \(\approx\) \(0.7430658896 + 0.01166850913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.797 + 0.603i)T \)
5 \( 1 + (-0.0249 - 0.999i)T \)
11 \( 1 + (-0.797 + 0.603i)T \)
13 \( 1 + (0.542 - 0.840i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (0.698 + 0.715i)T \)
29 \( 1 + (0.698 - 0.715i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.853 + 0.521i)T \)
43 \( 1 + (0.878 - 0.478i)T \)
47 \( 1 + (0.921 + 0.388i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.318 - 0.947i)T \)
61 \( 1 + (0.270 + 0.962i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.542 + 0.840i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−20.82648462492562994016851246301, −20.2645235784907427942313476281, −19.231067495964312104875526583616, −18.630464783374139380810054919959, −18.26790557020069798237017909442, −17.42308407913907094067320621319, −16.43277529276468516326899762126, −15.84391770576049966557274597768, −15.02953452814614991769446903052, −13.71894954075643693094069383354, −13.50319521690798906291458581779, −12.1998394462545258118063146487, −11.4433151703465675625520753713, −10.86710162810939215424483821296, −10.24641285265777707525119493260, −9.26542173701717125976147740846, −8.573727987821889518984158530509, −7.61536477604425676361859252755, −6.907780962173183012806180129419, −6.14565859733422741178608338441, −4.76565081454081142991258064017, −3.6775749110982082966855626639, −2.86194579423893410678362286656, −2.21208449600053939911828530254, −0.8771014845470023003136237843, 0.65437589816130574626243168827, 1.5942692446316762548144767408, 2.70542003208385434753367492472, 4.15057613089196990608754560028, 5.206603893086971132913661778452, 5.585734717011947292877666284357, 6.77106986760729891411224502825, 7.64451124195187987967548467525, 8.309797031906410499070496783581, 9.01366229937692011718836385222, 9.821096740829113694776771527366, 10.57678620313374100597267037402, 11.43168758141027024923174155066, 12.47133068991544572167592738993, 13.262847705483681645535985528854, 13.96565624184210664017894063514, 15.23388392208217248298980315654, 15.666562127989298726542286935419, 16.18953453216490207643353717521, 17.33908448876762283939361755950, 17.63369517031623559192763362847, 18.39375953949472207020680598031, 19.44020517269369854180374153706, 20.04452001045281917429264039858, 20.63366707237983034714127996124

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