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Properties

Label 1-72e2-5184.1019-r0-0-0
Degree $1$
Conductor $5184$
Sign $0.556 + 0.830i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
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  + (0.962 + 0.272i)5-s + (0.314 + 0.949i)7-s + (−0.999 + 0.0145i)11-s + (0.987 − 0.159i)13-s + (0.984 + 0.173i)17-s + (0.537 + 0.843i)19-s + (0.314 − 0.949i)23-s + (0.851 + 0.524i)25-s + (−0.994 + 0.101i)29-s + (−0.835 − 0.549i)31-s + (0.0436 + 0.999i)35-s + (0.999 + 0.0436i)37-s + (−0.0290 − 0.999i)41-s + (0.512 + 0.858i)43-s + (0.549 + 0.835i)47-s + ⋯
L(s)  = 1  + (0.962 + 0.272i)5-s + (0.314 + 0.949i)7-s + (−0.999 + 0.0145i)11-s + (0.987 − 0.159i)13-s + (0.984 + 0.173i)17-s + (0.537 + 0.843i)19-s + (0.314 − 0.949i)23-s + (0.851 + 0.524i)25-s + (−0.994 + 0.101i)29-s + (−0.835 − 0.549i)31-s + (0.0436 + 0.999i)35-s + (0.999 + 0.0436i)37-s + (−0.0290 − 0.999i)41-s + (0.512 + 0.858i)43-s + (0.549 + 0.835i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ 0.556 + 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.220777700 + 1.185300635i\)
\(L(\frac12)\) \(\approx\) \(2.220777700 + 1.185300635i\)
\(L(1)\) \(\approx\) \(1.370289089 + 0.2903151788i\)
\(L(1)\) \(\approx\) \(1.370289089 + 0.2903151788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.962 + 0.272i)T \)
7 \( 1 + (0.314 + 0.949i)T \)
11 \( 1 + (-0.999 + 0.0145i)T \)
13 \( 1 + (0.987 - 0.159i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (0.537 + 0.843i)T \)
23 \( 1 + (0.314 - 0.949i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
37 \( 1 + (0.999 + 0.0436i)T \)
41 \( 1 + (-0.0290 - 0.999i)T \)
43 \( 1 + (0.512 + 0.858i)T \)
47 \( 1 + (0.549 + 0.835i)T \)
53 \( 1 + (0.608 + 0.793i)T \)
59 \( 1 + (0.717 - 0.696i)T \)
61 \( 1 + (0.900 - 0.435i)T \)
67 \( 1 + (-0.775 + 0.631i)T \)
71 \( 1 + (-0.0871 - 0.996i)T \)
73 \( 1 + (-0.0871 + 0.996i)T \)
79 \( 1 + (-0.727 + 0.686i)T \)
83 \( 1 + (0.912 + 0.409i)T \)
89 \( 1 + (-0.996 - 0.0871i)T \)
97 \( 1 + (0.993 + 0.116i)T \)
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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

−17.91885233507532231969583416172, −17.28247863400227193622230263120, −16.4105812302972230808194609162, −16.234735727425329993244782288007, −15.15017083049246921704374494780, −14.49209479676949579675164533659, −13.64596856122461627447751221800, −13.35957220217331448063067770883, −12.86914510205263961004003380650, −11.72620714267761346362708523506, −11.095325419191382901974563890970, −10.44229045677035756321340973666, −9.860787086028699347158427937838, −9.14460045398208535812869245352, −8.4325080985699411272125112471, −7.49597901396263305946465396475, −7.15020399340464267085887693421, −6.05216778150792823707298064599, −5.42861783090021415427271250289, −4.93178491845189272166147088710, −3.89369565224122950209322696888, −3.20956029048120892924385442130, −2.268059433549407831562699460922, −1.39326957987610054419575832192, −0.75094591721156535631456318333, 1.01510647278930758954732864585, 1.85609236929866504082319271587, 2.57240888452808363445914005874, 3.21634691238069793591601796083, 4.194342471635519420522001847077, 5.3485700879260963482106339470, 5.6580141049936395314776630200, 6.134183426757615631445359788842, 7.2385835426216463712237744449, 7.95662482518257443864365927632, 8.59902080319182249545082050811, 9.39192767691282015252339407845, 9.943079667128131833609536341373, 10.79732923652783947361001511873, 11.16706605101816468178007818520, 12.24264150350187954344466995687, 12.80430176975642580055345977670, 13.33677454344438524809312645362, 14.27302672052190384616475522733, 14.64111121397645389530743071880, 15.38824589085861882129234867706, 16.18751321386297328424245413808, 16.67402655852410060910567082215, 17.589057516926371212925234462501, 18.25260915091879358484332622000

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