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Properties

Label 1-1323-1323.1202-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.989 + 0.143i$
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  + (−0.980 + 0.198i)2-s + (0.921 − 0.388i)4-s + (−0.969 + 0.246i)5-s + (−0.826 + 0.563i)8-s + (0.900 − 0.433i)10-s + (−0.980 + 0.198i)11-s + (0.853 − 0.521i)13-s + (0.698 − 0.715i)16-s + (−0.222 + 0.974i)17-s − 19-s + (−0.797 + 0.603i)20-s + (0.921 − 0.388i)22-s + (0.124 − 0.992i)23-s + (0.878 − 0.478i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (−0.980 + 0.198i)2-s + (0.921 − 0.388i)4-s + (−0.969 + 0.246i)5-s + (−0.826 + 0.563i)8-s + (0.900 − 0.433i)10-s + (−0.980 + 0.198i)11-s + (0.853 − 0.521i)13-s + (0.698 − 0.715i)16-s + (−0.222 + 0.974i)17-s − 19-s + (−0.797 + 0.603i)20-s + (0.921 − 0.388i)22-s + (0.124 − 0.992i)23-s + (0.878 − 0.478i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5842852517 + 0.04204246645i\)
\(L(\frac12)\) \(\approx\) \(0.5842852517 + 0.04204246645i\)
\(L(1)\) \(\approx\) \(0.5477401973 + 0.05252333135i\)
\(L(1)\) \(\approx\) \(0.5477401973 + 0.05252333135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.980 + 0.198i)T \)
5 \( 1 + (-0.969 + 0.246i)T \)
11 \( 1 + (-0.980 + 0.198i)T \)
13 \( 1 + (0.853 - 0.521i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.124 - 0.992i)T \)
29 \( 1 + (0.124 + 0.992i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.698 + 0.715i)T \)
43 \( 1 + (0.270 + 0.962i)T \)
47 \( 1 + (-0.661 - 0.749i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.995 - 0.0995i)T \)
61 \( 1 + (-0.921 - 0.388i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.853 - 0.521i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−20.93791694332340651595878220049, −20.058823626667953614883185900762, −19.155652575893075905068455299652, −18.87757413672297708228225157208, −17.963078361287221265727532582409, −17.201542650015207370257900083886, −16.24638321194615785972962949850, −15.76309396490994185965274099686, −15.28599405964388578353913785708, −13.97505333877415547304416064679, −13.02356827407566842851831998898, −12.223821484724213879622015559301, −11.38051577492697384929759443121, −10.93879234999281514213779147029, −10.01827050308452898417351871675, −8.96320764197123653091494814273, −8.47366797757524298464347048347, −7.62779892095996126417465927620, −6.979693753630802457937879218913, −5.95181855896842111307074373690, −4.783876701718878193445448490265, −3.73137621685451496086756887208, −2.91279570314248373833067428922, −1.819388675994978715044256619479, −0.61504375848394145376436937889, 0.5590831496014236110675297942, 1.91292547988543014145433738065, 2.889479019091222851219908925019, 3.85468891311321886901032043362, 4.99595582085372552376140239095, 6.13964188559732166441600240784, 6.77611396938825894300003863104, 7.89080125163474322019577400099, 8.19108077044980796901518588015, 9.000077772826642808818949142632, 10.23046141936666903663564177272, 10.809440811311902106710981266811, 11.24488379241504799357439410229, 12.5157315912217612331358257005, 12.93807808585941766433924053737, 14.48080184145169748754685147582, 15.04909326162475832800242368343, 15.7445666120861996349230270391, 16.31538603607324664322856037912, 17.186745577217313978793889046704, 18.13697624038018191615614317852, 18.556415795168499095620194363302, 19.378570794589664968050679512914, 19.98633848957922266502855795771, 20.76674819285730970999071850910

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