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Properties

Label 1-1323-1323.4-r0-0-0
Degree $1$
Conductor $1323$
Sign $-0.370 - 0.928i$
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  + (−0.318 + 0.947i)2-s + (−0.797 − 0.603i)4-s + (−0.969 + 0.246i)5-s + (0.826 − 0.563i)8-s + (0.0747 − 0.997i)10-s + (0.980 − 0.198i)11-s + (−0.0249 − 0.999i)13-s + (0.270 + 0.962i)16-s + (−0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.921 + 0.388i)20-s + (−0.124 + 0.992i)22-s + (−0.124 + 0.992i)23-s + (0.878 − 0.478i)25-s + (0.955 + 0.294i)26-s + ⋯
L(s)  = 1  + (−0.318 + 0.947i)2-s + (−0.797 − 0.603i)4-s + (−0.969 + 0.246i)5-s + (0.826 − 0.563i)8-s + (0.0747 − 0.997i)10-s + (0.980 − 0.198i)11-s + (−0.0249 − 0.999i)13-s + (0.270 + 0.962i)16-s + (−0.733 − 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.921 + 0.388i)20-s + (−0.124 + 0.992i)22-s + (−0.124 + 0.992i)23-s + (0.878 − 0.478i)25-s + (0.955 + 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.370 - 0.928i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ -0.370 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1040618284 - 0.1536083055i\)
\(L(\frac12)\) \(\approx\) \(0.1040618284 - 0.1536083055i\)
\(L(1)\) \(\approx\) \(0.5806285220 + 0.1780298447i\)
\(L(1)\) \(\approx\) \(0.5806285220 + 0.1780298447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.318 + 0.947i)T \)
5 \( 1 + (-0.969 + 0.246i)T \)
11 \( 1 + (0.980 - 0.198i)T \)
13 \( 1 + (-0.0249 - 0.999i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.124 + 0.992i)T \)
29 \( 1 + (-0.797 + 0.603i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.969 + 0.246i)T \)
43 \( 1 + (0.698 - 0.715i)T \)
47 \( 1 + (0.980 - 0.198i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (-0.583 - 0.811i)T \)
61 \( 1 + (-0.797 + 0.603i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.0249 + 0.999i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−20.96895339989192047535981995504, −20.38413430276385898820770944584, −19.45019696459125505680587926877, −19.23132304321399787683113918994, −18.40424331095142642038178464081, −17.33965368164783138299389743870, −16.78974985135735244258736337426, −16.04687030644388538841776659038, −14.896058071492097782898560440389, −14.26164287182006457818853925429, −13.28089864525463022692330721503, −12.255447666419414326262447447840, −12.066991774953267748751135733, −11.092554632940855196327559463589, −10.49254116341267365038510741846, −9.33722439332536746934892118982, −8.78952957175894021198686885569, −8.05096385704320992260980940922, −7.0983578517652585949524112819, −6.13009018071517275075723463857, −4.548817500448538871331773454875, −4.225166274600204397351986372094, −3.42390234994778713846103998850, −2.15001148778787146833215588013, −1.34389045079253593408154347285, 0.09229956728958769658130497372, 1.26898627711720897340510821895, 2.89163462203237921807121386695, 3.9127320117128590368456608296, 4.65917850403578872670260641940, 5.62818333686136290133187693339, 6.633287637889831018635271665917, 7.215242520283448090481788947234, 8.020439056348020881445042936004, 8.803342638144920010968010547430, 9.489392650752051647846220797394, 10.586295773368299089830640250, 11.29322319329536160903796489020, 12.179022257248059119121118319810, 13.25550267386413321318344459449, 13.89715229232268326405669615576, 14.9369060979074591807731344661, 15.384438031206788886218211406632, 15.935196370008704104951933261172, 17.02670944088292911097345549857, 17.42765021164981363891995412723, 18.449228827318848933321777481979, 19.03043218767259148679433093160, 19.88955460117696102801072667924, 20.28154847542556632151018525746

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