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Properties

Label 1-4729-4729.4438-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.636 - 0.770i$
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  + (0.978 − 0.205i)2-s + (0.639 + 0.768i)3-s + (0.915 − 0.402i)4-s + (0.999 + 0.00531i)5-s + (0.783 + 0.620i)6-s + (−0.996 − 0.0849i)7-s + (0.812 − 0.582i)8-s + (−0.182 + 0.983i)9-s + (0.979 − 0.200i)10-s + (−0.998 + 0.0557i)11-s + (0.894 + 0.446i)12-s + (0.567 − 0.823i)13-s + (−0.992 + 0.121i)14-s + (0.635 + 0.772i)15-s + (0.675 − 0.737i)16-s + (0.782 − 0.622i)17-s + ⋯
L(s)  = 1  + (0.978 − 0.205i)2-s + (0.639 + 0.768i)3-s + (0.915 − 0.402i)4-s + (0.999 + 0.00531i)5-s + (0.783 + 0.620i)6-s + (−0.996 − 0.0849i)7-s + (0.812 − 0.582i)8-s + (−0.182 + 0.983i)9-s + (0.979 − 0.200i)10-s + (−0.998 + 0.0557i)11-s + (0.894 + 0.446i)12-s + (0.567 − 0.823i)13-s + (−0.992 + 0.121i)14-s + (0.635 + 0.772i)15-s + (0.675 − 0.737i)16-s + (0.782 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.636 - 0.770i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (4438, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.636 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.099092172 - 1.930371937i\)
\(L(\frac12)\) \(\approx\) \(4.099092172 - 1.930371937i\)
\(L(1)\) \(\approx\) \(2.440447354 - 0.2467915996i\)
\(L(1)\) \(\approx\) \(2.440447354 - 0.2467915996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.978 - 0.205i)T \)
3 \( 1 + (0.639 + 0.768i)T \)
5 \( 1 + (0.999 + 0.00531i)T \)
7 \( 1 + (-0.996 - 0.0849i)T \)
11 \( 1 + (-0.998 + 0.0557i)T \)
13 \( 1 + (0.567 - 0.823i)T \)
17 \( 1 + (0.782 - 0.622i)T \)
19 \( 1 + (-0.405 - 0.914i)T \)
23 \( 1 + (0.174 - 0.984i)T \)
29 \( 1 + (0.205 - 0.978i)T \)
31 \( 1 + (-0.358 + 0.933i)T \)
37 \( 1 + (-0.0345 - 0.999i)T \)
41 \( 1 + (-0.298 - 0.954i)T \)
43 \( 1 + (-0.966 - 0.254i)T \)
47 \( 1 + (0.573 - 0.818i)T \)
53 \( 1 + (-0.959 + 0.280i)T \)
59 \( 1 + (-0.845 + 0.534i)T \)
61 \( 1 + (0.483 - 0.875i)T \)
67 \( 1 + (-0.999 - 0.0265i)T \)
71 \( 1 + (0.937 - 0.348i)T \)
73 \( 1 + (0.991 + 0.132i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.618 + 0.785i)T \)
89 \( 1 + (-0.733 - 0.679i)T \)
97 \( 1 + (0.996 + 0.0849i)T \)
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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

−18.51306323727855266131595627597, −17.50957027386601665244508822727, −16.71338020450195897395246727977, −16.29508056778975247324509089644, −15.33302835853788370806435438583, −14.776391160264528999990988810142, −14.00570730166844832026381499176, −13.55175305497879402153369754945, −12.95054599402444172319571345765, −12.62742966296633491999853787805, −11.81141033319177215813158105938, −10.868891145104751344554323418087, −10.04016945101464165079988346771, −9.44579420633990468783045410988, −8.47750205126100395145487538849, −7.85681379524486692827304868933, −7.01898784182800530272916357726, −6.31863248837493530362664683240, −5.96296284397788460250735335918, −5.21908246582811147395234897397, −4.077222432195656130978053144358, −3.20624009855371350253108068011, −2.893176285004522157187378799516, −1.80427998358588048099922950842, −1.436263131102258836598074059856, 0.66453279523095219651054312368, 2.05073976458909019085817806939, 2.66967427627824132642893333377, 3.13620157293143302470897091585, 3.87674406532634474506286530467, 4.943977069081573125052176997192, 5.310136495865498719218288186159, 6.06643622203774525582475729722, 6.84549433981648799779310568729, 7.65832021887016315523669196858, 8.62879074514211481754642580564, 9.393207792207957779578560085884, 10.15464729143551130963916149588, 10.47822232360697781869844949682, 11.04300601694595718331883840475, 12.357345200151395772155337065206, 12.853627974569444939728661229825, 13.51937881450794524954232500079, 13.85712403800127608860437994536, 14.621508876680514310135156775361, 15.53283659745146266547998729188, 15.69376607826072085423839135945, 16.53380237671971562214863533281, 17.09064643809936002380499121170, 18.265921090291416151448677204147

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