Properties

Label 1-4729-4729.687-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.999 + 0.0417i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.0557 − 0.998i)2-s + (−0.244 − 0.969i)3-s + (−0.993 + 0.111i)4-s + (0.601 − 0.798i)5-s + (−0.954 + 0.298i)6-s + (−0.614 − 0.788i)7-s + (0.166 + 0.986i)8-s + (−0.880 + 0.474i)9-s + (−0.830 − 0.556i)10-s + (−0.959 − 0.283i)11-s + (0.351 + 0.936i)12-s + (0.260 − 0.965i)13-s + (−0.753 + 0.657i)14-s + (−0.921 − 0.388i)15-s + (0.975 − 0.221i)16-s + (−0.721 − 0.692i)17-s + ⋯
L(s)  = 1  + (−0.0557 − 0.998i)2-s + (−0.244 − 0.969i)3-s + (−0.993 + 0.111i)4-s + (0.601 − 0.798i)5-s + (−0.954 + 0.298i)6-s + (−0.614 − 0.788i)7-s + (0.166 + 0.986i)8-s + (−0.880 + 0.474i)9-s + (−0.830 − 0.556i)10-s + (−0.959 − 0.283i)11-s + (0.351 + 0.936i)12-s + (0.260 − 0.965i)13-s + (−0.753 + 0.657i)14-s + (−0.921 − 0.388i)15-s + (0.975 − 0.221i)16-s + (−0.721 − 0.692i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.999 + 0.0417i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.999 + 0.0417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4975096188 + 0.01038260285i\)
\(L(\frac12)\) \(\approx\) \(-0.4975096188 + 0.01038260285i\)
\(L(1)\) \(\approx\) \(0.2235415694 - 0.6614106601i\)
\(L(1)\) \(\approx\) \(0.2235415694 - 0.6614106601i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.0557 - 0.998i)T \)
3 \( 1 + (-0.244 - 0.969i)T \)
5 \( 1 + (0.601 - 0.798i)T \)
7 \( 1 + (-0.614 - 0.788i)T \)
11 \( 1 + (-0.959 - 0.283i)T \)
13 \( 1 + (0.260 - 0.965i)T \)
17 \( 1 + (-0.721 - 0.692i)T \)
19 \( 1 + (0.939 + 0.343i)T \)
23 \( 1 + (-0.627 - 0.778i)T \)
29 \( 1 + (0.0557 - 0.998i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
37 \( 1 + (-0.963 + 0.267i)T \)
41 \( 1 + (0.773 - 0.633i)T \)
43 \( 1 + (0.639 - 0.768i)T \)
47 \( 1 + (-0.901 - 0.431i)T \)
53 \( 1 + (0.709 - 0.704i)T \)
59 \( 1 + (-0.793 - 0.608i)T \)
61 \( 1 + (-0.995 - 0.0955i)T \)
67 \( 1 + (-0.0875 + 0.996i)T \)
71 \( 1 + (0.651 - 0.758i)T \)
73 \( 1 + (-0.424 + 0.905i)T \)
79 \( 1 - T \)
83 \( 1 + (0.987 - 0.158i)T \)
89 \( 1 + (-0.410 + 0.912i)T \)
97 \( 1 + (-0.614 - 0.788i)T \)
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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

−18.48874920354116371474370511525, −17.98217262024952183816579497550, −17.56364909327491619052670650896, −16.552538010802069956101187023847, −16.11998701133204471820638346375, −15.45017825893844241314193092944, −15.08908442150268213104920183, −14.30419062165688319468744111617, −13.686662756614073256883910918431, −13.0011445186720494811450257258, −12.12351377198817219581598497241, −11.1446007040905149839728193390, −10.57172861822057232595732030384, −9.697968103696442230867088964208, −9.37706851756257163399071963110, −8.78367669945481838481653492611, −7.76266255467648423393423632679, −6.98498620808735748496750494047, −6.16485907641684298452145148790, −5.81333055212488527754063714202, −5.10366423146654294928200684810, −4.29198067989128207973946684883, −3.42992938244498109997496543878, −2.75543975571972812461309215733, −1.67750545362598540589080372036, 0.19727765433668456777760229269, 0.70138010077951960218986004138, 1.59110817653342097288718819082, 2.4456431295716085282799260058, 3.0532369891937667276967398093, 4.0007577128557503448054442549, 5.0507563298412906127019661930, 5.476671549418218052214643093613, 6.26764850495216504957253637683, 7.29494779470134454828819796516, 8.002885758138663579245195591502, 8.58772918668282787557774784386, 9.393387578033222165516308399994, 10.24365933069334388241097913545, 10.60217173973656505651805111180, 11.47337880419782512458009030567, 12.34957275605382103886473220213, 12.6848429525885955129560987665, 13.457761541445579643742682397812, 13.64996425426461577756249567963, 14.28697699481326414958767704606, 15.75861568489497231856538142839, 16.30782311128858945139773385681, 17.10687006819086423968352015275, 17.74600223925866969065505974592

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