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Properties

Label 1-5328-5328.4939-r0-0-0
Degree $1$
Conductor $5328$
Sign $-0.454 - 0.890i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(24.7431\)
Root analytic conductor: \(24.7431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (4939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (0:\ ),\ -0.454 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2598677063 - 0.4243149607i\)
\(L(\frac12)\) \(\approx\) \(0.2598677063 - 0.4243149607i\)
\(L(1)\) \(\approx\) \(0.7482015859 - 0.03271396335i\)
\(L(1)\) \(\approx\) \(0.7482015859 - 0.03271396335i\)

Euler product

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

−18.299955390117371393391497991413, −17.62892356501070474457856608955, −16.69038253652254587456587002701, −15.99994071461969820903639278524, −15.49977792847037320789117688582, −15.07397234920894790696791258835, −14.32301508268318336208665545119, −13.32807331308090485128050949564, −12.72911972189557791985017412095, −12.10336812525752482359348704159, −11.663433698973391155271361270152, −10.8770901410828094869321321902, −9.98389968244644464250905533950, −9.49653387551011427850864181474, −8.40522605610300343796022774939, −8.05855730595360819636103428096, −7.5429929789634058045084901759, −6.532886640365263254056082369916, −5.62628353329591340006020338399, −5.10665053835123020739225328191, −4.39424495742171398673495052834, −3.534163256588577439369135151892, −2.64713181112510509582598150829, −2.11812435170712675003533899518, −0.80896198648536582893475882376, 0.17135467536916919773351084544, 1.24077704734483509652335223099, 2.247347315956587643679849140964, 3.16131526659227399959254789090, 3.86320540658795353551289074863, 4.443587854222307768622541546294, 5.17110755829596594178964871391, 6.24657628349597548705764540979, 6.94270881816043041195604282119, 7.47692861814407630377599349833, 8.18049400319855165808419247081, 8.76242885391015617908951367256, 9.762194804102760672475868500816, 10.50513896221262500009575424444, 11.086904131412504399241219796080, 11.48518381789504481256031803638, 12.40905752501346665054693932856, 13.084344487620222282195775860875, 13.86316841607763090098653882810, 14.29937345254092381426657636959, 15.164877742832852088082360360759, 15.79268339740468581498216357138, 16.30356544679318491517713557214, 17.00115234891858186694021864962, 17.778285376071164438054078695846

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