↑

Properties

Label 1-4729-4729.1080-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.0354 + 0.999i$
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  + (−0.380 − 0.924i)2-s + (−0.987 + 0.158i)3-s + (−0.709 + 0.704i)4-s + (0.326 − 0.945i)5-s + (0.522 + 0.852i)6-s + (−0.567 + 0.823i)7-s + (0.921 + 0.388i)8-s + (0.949 − 0.313i)9-s + (−0.998 + 0.0584i)10-s + (−0.905 − 0.424i)11-s + (0.589 − 0.808i)12-s + (0.713 − 0.700i)13-s + (0.977 + 0.211i)14-s + (−0.171 + 0.985i)15-s + (0.00797 − 0.999i)16-s + (0.920 − 0.390i)17-s + ⋯
L(s)  = 1  + (−0.380 − 0.924i)2-s + (−0.987 + 0.158i)3-s + (−0.709 + 0.704i)4-s + (0.326 − 0.945i)5-s + (0.522 + 0.852i)6-s + (−0.567 + 0.823i)7-s + (0.921 + 0.388i)8-s + (0.949 − 0.313i)9-s + (−0.998 + 0.0584i)10-s + (−0.905 − 0.424i)11-s + (0.589 − 0.808i)12-s + (0.713 − 0.700i)13-s + (0.977 + 0.211i)14-s + (−0.171 + 0.985i)15-s + (0.00797 − 0.999i)16-s + (0.920 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.0354 + 0.999i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1080, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.0354 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1549211994 - 0.1605084677i\)
\(L(\frac12)\) \(\approx\) \(-0.1549211994 - 0.1605084677i\)
\(L(1)\) \(\approx\) \(0.4247371802 - 0.3157827772i\)
\(L(1)\) \(\approx\) \(0.4247371802 - 0.3157827772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.380 - 0.924i)T \)
3 \( 1 + (-0.987 + 0.158i)T \)
5 \( 1 + (0.326 - 0.945i)T \)
7 \( 1 + (-0.567 + 0.823i)T \)
11 \( 1 + (-0.905 - 0.424i)T \)
13 \( 1 + (0.713 - 0.700i)T \)
17 \( 1 + (0.920 - 0.390i)T \)
19 \( 1 + (-0.161 - 0.986i)T \)
23 \( 1 + (-0.0318 - 0.999i)T \)
29 \( 1 + (-0.924 - 0.380i)T \)
31 \( 1 + (0.595 + 0.803i)T \)
37 \( 1 + (-0.980 - 0.195i)T \)
41 \( 1 + (-0.996 + 0.0875i)T \)
43 \( 1 + (0.928 - 0.370i)T \)
47 \( 1 + (-0.873 - 0.486i)T \)
53 \( 1 + (0.958 + 0.285i)T \)
59 \( 1 + (-0.790 + 0.612i)T \)
61 \( 1 + (-0.989 - 0.145i)T \)
67 \( 1 + (-0.995 - 0.0902i)T \)
71 \( 1 + (0.265 + 0.964i)T \)
73 \( 1 + (0.899 + 0.436i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.997 + 0.0690i)T \)
89 \( 1 + (0.333 - 0.942i)T \)
97 \( 1 + (0.567 - 0.823i)T \)
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

−18.50633950604143505841069160604, −17.96982696267071702758271380293, −17.16079894888705195856346664727, −16.78732712823805760747376809861, −16.110747030531676586855997698855, −15.47103864521900111257523753335, −14.80732943390094383797085031222, −13.876783214650970881467675461555, −13.49391411555009477382151200969, −12.78732657220333827188124427622, −11.84821262936735088475820904094, −10.8286655946302024857651358254, −10.52824502335808411132379611359, −9.87222076901005475000617910744, −9.36663510999991676078208548155, −7.894930075471363519754050428423, −7.625902758104195380917378307628, −6.84717400070002155439641050519, −6.27565733753927035976011877996, −5.75641524505720649511199339487, −5.01852587343794336932502523142, −3.99083158458793192396525910169, −3.43592938661731924622663439794, −1.87937859979399012458257119005, −1.277040014751343576551411689899, 0.11121470121451476305128056341, 0.780862845259276417617720507881, 1.71203742455402338580169931929, 2.69261394280468621059192054576, 3.3931679357264118282987058012, 4.38597675461407233189075284052, 5.24107449063005105581258425196, 5.48368744208468516592893062286, 6.391227009349653986502483570569, 7.466393684430901765070414261037, 8.37977286997236564455972979190, 8.87632886449761714614025838644, 9.601451273383240998985971069739, 10.347119652742130233387432688803, 10.718819151952925778617328096673, 11.740800642339580300634728138401, 12.12551110830618211548051204491, 12.89915101717413074460870313489, 13.13784840364778228054815468989, 13.95261878157377893044816101246, 15.38092335974767676570619215616, 15.8129975698695909875739598645, 16.55366906582086122674677534819, 16.96825484671747163353596916264, 17.85232051369678961214180513783

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