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Properties

Label 1-4729-4729.65-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: $\chi_{4729} (65, \cdot )$
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

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

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