Properties

Label 1-4729-4729.4689-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.995 - 0.0955i$
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  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (−0.295 + 0.955i)5-s + (−0.989 − 0.143i)6-s + (0.0928 − 0.995i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.405 − 0.914i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (0.311 − 0.950i)13-s + (0.584 + 0.811i)14-s + (−0.917 + 0.397i)15-s + (−0.963 − 0.267i)16-s + (0.472 − 0.881i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (−0.295 + 0.955i)5-s + (−0.989 − 0.143i)6-s + (0.0928 − 0.995i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.405 − 0.914i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (0.311 − 0.950i)13-s + (0.584 + 0.811i)14-s + (−0.917 + 0.397i)15-s + (−0.963 − 0.267i)16-s + (0.472 − 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.995 - 0.0955i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (4689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.995 - 0.0955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8257946658 - 0.03954436846i\)
\(L(\frac12)\) \(\approx\) \(0.8257946658 - 0.03954436846i\)
\(L(1)\) \(\approx\) \(0.6953615419 + 0.3466594628i\)
\(L(1)\) \(\approx\) \(0.6953615419 + 0.3466594628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.753 + 0.657i)T \)
3 \( 1 + (0.651 + 0.758i)T \)
5 \( 1 + (-0.295 + 0.955i)T \)
7 \( 1 + (0.0928 - 0.995i)T \)
11 \( 1 + (-0.698 + 0.715i)T \)
13 \( 1 + (0.311 - 0.950i)T \)
17 \( 1 + (0.472 - 0.881i)T \)
19 \( 1 + (-0.717 - 0.696i)T \)
23 \( 1 + (-0.467 + 0.884i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.954 + 0.298i)T \)
37 \( 1 + (0.919 + 0.393i)T \)
41 \( 1 + (-0.987 - 0.158i)T \)
43 \( 1 + (0.937 - 0.348i)T \)
47 \( 1 + (-0.0186 - 0.999i)T \)
53 \( 1 + (-0.910 - 0.412i)T \)
59 \( 1 + (-0.913 + 0.407i)T \)
61 \( 1 + (0.254 - 0.966i)T \)
67 \( 1 + (-0.997 + 0.0690i)T \)
71 \( 1 + (0.956 + 0.293i)T \)
73 \( 1 + (-0.940 + 0.338i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.702 - 0.711i)T \)
89 \( 1 + (-0.540 + 0.841i)T \)
97 \( 1 + (0.0928 - 0.995i)T \)
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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

−18.46943710585923593317028174505, −17.64605191754119115271212893171, −16.96571517275520599532824863509, −16.237411816790678919522586103544, −15.7049662778241881192801727546, −14.82264628166206172865848093191, −13.9429452213813065552422589093, −13.2226519863060519784855026372, −12.59545983030166074249478631382, −12.162138656445291461993680322028, −11.6217415445003646665048999259, −10.73993712391817106308322924528, −9.80203144743003156594580441240, −9.110839042097543900570594788003, −8.5105552064796145729331798759, −8.1383603882065010389049046594, −7.6843494953090700586930488670, −6.23476498213806933648674093175, −6.095692164885182117273045902110, −4.60465820281519767268578020298, −3.98817433945463660432090746504, −3.03131220107511710279828662817, −2.343915368371534373171242818205, −1.6528499265203745374717933105, −0.91565447496487405948160590317, 0.29803797565022784550609935280, 1.58317328137220049073172785861, 2.648655665648755089014443912143, 3.16108238155179183794351293253, 4.225676193226206405974662322817, 4.86958831516114615879176792651, 5.64289656247827150000676067909, 6.76160674671906721840187317008, 7.22654446505747717278733704574, 7.99116427991297188098266312997, 8.25673313162712391711441229589, 9.46047614840410909358012403100, 9.97896261800975227668568703275, 10.525693858572274724617225625962, 10.88874861824491040834066008797, 11.76953816414754170656524643893, 13.11992422422643882053631307441, 13.79805370637919657229159831443, 14.27013470324022704308530308956, 15.01933005506133554876451697568, 15.60480277570787306602352637392, 15.843130645797627482806597849930, 16.81151918598361143266535812808, 17.513057041278924956874311852664, 18.05685709536911796083745074885

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