Properties

Label 1-4729-4729.3610-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.999 + 0.00863i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.933 + 0.358i)2-s + (−0.481 − 0.876i)3-s + (0.742 − 0.669i)4-s + (−0.793 − 0.608i)5-s + (0.763 + 0.645i)6-s + (−0.509 − 0.860i)7-s + (−0.453 + 0.891i)8-s + (−0.536 + 0.843i)9-s + (0.959 + 0.283i)10-s + (−0.549 − 0.835i)11-s + (−0.944 − 0.328i)12-s + (0.927 + 0.373i)13-s + (0.783 + 0.620i)14-s + (−0.150 + 0.988i)15-s + (0.103 − 0.994i)16-s + (−0.856 − 0.516i)17-s + ⋯
L(s)  = 1  + (−0.933 + 0.358i)2-s + (−0.481 − 0.876i)3-s + (0.742 − 0.669i)4-s + (−0.793 − 0.608i)5-s + (0.763 + 0.645i)6-s + (−0.509 − 0.860i)7-s + (−0.453 + 0.891i)8-s + (−0.536 + 0.843i)9-s + (0.959 + 0.283i)10-s + (−0.549 − 0.835i)11-s + (−0.944 − 0.328i)12-s + (0.927 + 0.373i)13-s + (0.783 + 0.620i)14-s + (−0.150 + 0.988i)15-s + (0.103 − 0.994i)16-s + (−0.856 − 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.999 + 0.00863i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.999 + 0.00863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001574051935 - 0.3647169066i\)
\(L(\frac12)\) \(\approx\) \(0.001574051935 - 0.3647169066i\)
\(L(1)\) \(\approx\) \(0.4228972318 - 0.1857310629i\)
\(L(1)\) \(\approx\) \(0.4228972318 - 0.1857310629i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.933 + 0.358i)T \)
3 \( 1 + (-0.481 - 0.876i)T \)
5 \( 1 + (-0.793 - 0.608i)T \)
7 \( 1 + (-0.509 - 0.860i)T \)
11 \( 1 + (-0.549 - 0.835i)T \)
13 \( 1 + (0.927 + 0.373i)T \)
17 \( 1 + (-0.856 - 0.516i)T \)
19 \( 1 + (0.872 + 0.488i)T \)
23 \( 1 + (-0.915 + 0.402i)T \)
29 \( 1 + (0.933 + 0.358i)T \)
31 \( 1 + (0.424 + 0.905i)T \)
37 \( 1 + (-0.894 - 0.446i)T \)
41 \( 1 + (0.908 + 0.417i)T \)
43 \( 1 + (-0.290 + 0.956i)T \)
47 \( 1 + (0.978 - 0.205i)T \)
53 \( 1 + (-0.410 - 0.912i)T \)
59 \( 1 + (0.981 - 0.190i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + (0.991 + 0.127i)T \)
71 \( 1 + (-0.984 - 0.174i)T \)
73 \( 1 + (0.803 + 0.595i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.366 - 0.930i)T \)
89 \( 1 + (0.687 - 0.726i)T \)
97 \( 1 + (-0.509 - 0.860i)T \)
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

−18.50437729046870601654745664302, −17.72005006592069301919605342795, −17.49470299370348003488076868227, −16.219308647819324426117236071012, −15.86181086882390673426409024996, −15.42214770259045079683113461070, −15.04794714743106043102344662027, −13.786037311605070792304362352686, −12.746931853108036434565421263125, −12.0857810332615807742015280199, −11.65697801160929578484342018254, −10.88937000475894932829570715633, −10.34855339138749208384075276847, −9.82966306667556699322787579469, −8.95321640044889209437514424933, −8.441058029926481384484873170176, −7.664179135886839709426508610697, −6.71256887396384087987427257432, −6.23105259082075340596875741119, −5.34084132345279501825130935535, −4.19048989249832716605198983029, −3.73267767785232851099361802045, −2.747310346719852386919510026437, −2.35086572913637434729309126134, −0.8196656282781665200837647932, 0.232415056671864920542876920386, 0.95221836046601064483507319568, 1.5872282635350489663703147657, 2.802598443901412922043851724883, 3.61987506603624635377636303617, 4.72414211809271817999246416858, 5.53947708632580870930462842093, 6.24765506407829405965590628641, 6.96541986421377590525126922230, 7.45668050259584226295661300968, 8.3523368741164323395276239379, 8.50662312663864797778715204009, 9.5967706699513664490249679100, 10.42167221006471921605770728255, 11.17105733251913223663762326487, 11.49483840600003187205488146976, 12.324410386004225397195306742830, 13.11377903434553917074249450720, 13.85283999622974047910561664988, 14.25567089755124163505644545679, 15.84651506342313348146183168466, 16.02580171512154574629188138233, 16.27928291643829576171007352551, 17.260494688296160198153554720018, 17.825847689564153757649168240956

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