Properties

Label 1-4729-4729.336-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.921 + 0.388i$
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  + (0.991 + 0.127i)2-s + (−0.709 + 0.704i)3-s + (0.967 + 0.252i)4-s + (−0.746 + 0.665i)5-s + (−0.793 + 0.608i)6-s + (0.610 + 0.792i)7-s + (0.927 + 0.373i)8-s + (0.00797 − 0.999i)9-s + (−0.824 + 0.565i)10-s + (−0.978 + 0.205i)11-s + (−0.864 + 0.502i)12-s + (0.827 + 0.560i)13-s + (0.504 + 0.863i)14-s + (0.0610 − 0.998i)15-s + (0.872 + 0.488i)16-s + (0.851 − 0.525i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.127i)2-s + (−0.709 + 0.704i)3-s + (0.967 + 0.252i)4-s + (−0.746 + 0.665i)5-s + (−0.793 + 0.608i)6-s + (0.610 + 0.792i)7-s + (0.927 + 0.373i)8-s + (0.00797 − 0.999i)9-s + (−0.824 + 0.565i)10-s + (−0.978 + 0.205i)11-s + (−0.864 + 0.502i)12-s + (0.827 + 0.560i)13-s + (0.504 + 0.863i)14-s + (0.0610 − 0.998i)15-s + (0.872 + 0.488i)16-s + (0.851 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.921 + 0.388i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (336, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.921 + 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4865371792 + 2.406007513i\)
\(L(\frac12)\) \(\approx\) \(0.4865371792 + 2.406007513i\)
\(L(1)\) \(\approx\) \(1.216719150 + 0.9050204192i\)
\(L(1)\) \(\approx\) \(1.216719150 + 0.9050204192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.991 + 0.127i)T \)
3 \( 1 + (-0.709 + 0.704i)T \)
5 \( 1 + (-0.746 + 0.665i)T \)
7 \( 1 + (0.610 + 0.792i)T \)
11 \( 1 + (-0.978 + 0.205i)T \)
13 \( 1 + (0.827 + 0.560i)T \)
17 \( 1 + (0.851 - 0.525i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
23 \( 1 + (-0.453 - 0.891i)T \)
29 \( 1 + (0.991 - 0.127i)T \)
31 \( 1 + (-0.999 - 0.0159i)T \)
37 \( 1 + (0.999 + 0.0212i)T \)
41 \( 1 + (0.783 + 0.620i)T \)
43 \( 1 + (0.690 + 0.722i)T \)
47 \( 1 + (0.476 + 0.879i)T \)
53 \( 1 + (0.952 - 0.303i)T \)
59 \( 1 + (-0.224 + 0.974i)T \)
61 \( 1 + (-0.997 + 0.0690i)T \)
67 \( 1 + (0.877 - 0.479i)T \)
71 \( 1 + (-0.0981 + 0.995i)T \)
73 \( 1 + (-0.800 - 0.599i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.285 + 0.958i)T \)
89 \( 1 + (-0.976 + 0.216i)T \)
97 \( 1 + (0.610 + 0.792i)T \)
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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

−17.79923545439474144861296132485, −17.04534285003212806272438108341, −16.497194111261060600666787299052, −15.769654989265853889057958301336, −15.35156002465741859743323193871, −14.29043296955612099388861878106, −13.64810335331652729279823971342, −13.012756890606504648053778913563, −12.65732235669438324496718867409, −11.86968090513505789633862407527, −11.19103667372370076370275645663, −10.76615918120108301074900373441, −10.180162855221037711563029860, −8.66099144858831343583433677558, −7.86788751710570121837428280245, −7.56674527657303399747146600026, −6.80545499092291590992247454320, −5.625252079477998254643198408907, −5.540931634590190763426798318024, −4.554216672238069046347906335287, −3.99740125025908817185080859101, −3.123137100736363199177983642850, −2.064367340655170446148065032485, −1.20963609636838996426052888917, −0.56048638965176503113549212046, 1.1901793492219059041246365848, 2.45409731316860484641351622982, 2.95902469030439537302717577701, 3.99799201865806899514290353477, 4.38810586622542686714437998853, 5.162646677500579467080895024916, 6.01105265590111393761222161608, 6.27132203484591037120349074997, 7.40525953010896896505133516954, 7.96319352886200325388838056558, 8.760864182713788080537590882081, 9.931368631343124064037577771548, 10.65498555869526850108955077849, 11.09604426513348863653118051386, 11.73318174598570124457456496753, 12.29863537769397176476437177815, 12.82713827449309603251770628537, 14.078088084706908332111168284583, 14.56546329814311539880623594369, 15.04881501416462709238009356995, 15.81700024383672310626492842450, 16.16565320832654749131556733680, 16.71718707656180805801300850598, 17.92736033386005945814867837927, 18.343161773417761063635733208362

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