↑

Properties

Label 1-5328-5328.493-r0-0-0
Degree $1$
Conductor $5328$
Sign $-0.320 + 0.947i$
Analytic cond. $24.7431$
Root an. cond. $24.7431$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)5-s + (0.939 + 0.342i)7-s − i·11-s + (0.642 + 0.766i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s − 23-s + (0.939 − 0.342i)25-s + i·29-s + (−0.5 + 0.866i)31-s + (0.984 + 0.173i)35-s + (−0.766 + 0.642i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (0.766 + 0.642i)49-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (0.939 + 0.342i)7-s − i·11-s + (0.642 + 0.766i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s − 23-s + (0.939 − 0.342i)25-s + i·29-s + (−0.5 + 0.866i)31-s + (0.984 + 0.173i)35-s + (−0.766 + 0.642i)41-s + (−0.866 + 0.5i)43-s + (−0.5 + 0.866i)47-s + (0.766 + 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.320 + 0.947i$
Analytic conductor: \(24.7431\)
Root analytic conductor: \(24.7431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (0:\ ),\ -0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8858385642 + 1.235433353i\)
\(L(\frac12)\) \(\approx\) \(0.8858385642 + 1.235433353i\)
\(L(1)\) \(\approx\) \(1.186431319 + 0.1784565978i\)
\(L(1)\) \(\approx\) \(1.186431319 + 0.1784565978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 - T \)
29 \( 1 + iT \)
31 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.642 + 0.766i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.807855853038287755435694722673, −17.19351182709147668949698752878, −16.6726846366591042734562961816, −15.41342883963751499095444770176, −15.21891018740661394277034524693, −14.40113157341375185911541948418, −13.654882676698482191263702768310, −13.26770167458461362858645386805, −12.51835716555732945073187303394, −11.59900287188520096905609377508, −11.0130051647179781938272738501, −10.20548738877019812271490666260, −9.92976209713533541505759868406, −8.835386990623098909384715484760, −8.381476104471846433160752165359, −7.4590647701717415483582457731, −6.84478110378343920336157303716, −6.01700415527513616118196938040, −5.421598577750961450341771037, −4.526403936065891202576743047782, −4.06194234075563307855673515861, −2.864035945735933319853196473859, −1.91633964667538796923613217399, −1.76440306296840585297303604443, −0.33222759782964752170409211809, 1.378240673529856065582006119, 1.70088277988395890600124046911, 2.55886121391897692995643748069, 3.52238115496837448932052631853, 4.45545288124630555250599432587, 5.01526235735865499998828685745, 6.01864391778933650382622885249, 6.22557278929706780091463072392, 7.156561199539413731033058519380, 8.34439141477435352833617864441, 8.63274343191236966005421363184, 9.14953816656636972012421675784, 10.236075954646969479286565548081, 10.756837537067240637311966009083, 11.39187860521520085825018184598, 12.06749568969233840833212184405, 13.038396018789411603710841353795, 13.44826032232709312417360490400, 14.27071803541749877085444425371, 14.57008738113327543824768352635, 15.485652970914448036035548969402, 16.40287844331335422351741274471, 16.674299791736878376961160026630, 17.62679067270267381087347849415, 18.112522992978394001566736906950

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