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Properties

Label 1-4729-4729.1081-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.904 - 0.425i$
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  + (0.424 + 0.905i)2-s + (0.864 + 0.502i)3-s + (−0.639 + 0.768i)4-s + (−0.977 − 0.211i)5-s + (−0.0875 + 0.996i)6-s + (0.966 + 0.257i)7-s + (−0.967 − 0.252i)8-s + (0.495 + 0.868i)9-s + (−0.224 − 0.974i)10-s + (−0.788 + 0.614i)11-s + (−0.939 + 0.343i)12-s + (0.540 − 0.841i)13-s + (0.177 + 0.984i)14-s + (−0.739 − 0.673i)15-s + (−0.182 − 0.983i)16-s + (0.981 − 0.192i)17-s + ⋯
L(s)  = 1  + (0.424 + 0.905i)2-s + (0.864 + 0.502i)3-s + (−0.639 + 0.768i)4-s + (−0.977 − 0.211i)5-s + (−0.0875 + 0.996i)6-s + (0.966 + 0.257i)7-s + (−0.967 − 0.252i)8-s + (0.495 + 0.868i)9-s + (−0.224 − 0.974i)10-s + (−0.788 + 0.614i)11-s + (−0.939 + 0.343i)12-s + (0.540 − 0.841i)13-s + (0.177 + 0.984i)14-s + (−0.739 − 0.673i)15-s + (−0.182 − 0.983i)16-s + (0.981 − 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.904 - 0.425i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.904 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4393941229 + 1.966330459i\)
\(L(\frac12)\) \(\approx\) \(-0.4393941229 + 1.966330459i\)
\(L(1)\) \(\approx\) \(0.9148686908 + 1.061568473i\)
\(L(1)\) \(\approx\) \(0.9148686908 + 1.061568473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.424 + 0.905i)T \)
3 \( 1 + (0.864 + 0.502i)T \)
5 \( 1 + (-0.977 - 0.211i)T \)
7 \( 1 + (0.966 + 0.257i)T \)
11 \( 1 + (-0.788 + 0.614i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (0.981 - 0.192i)T \)
19 \( 1 + (-0.554 + 0.832i)T \)
23 \( 1 + (-0.669 + 0.742i)T \)
29 \( 1 + (-0.905 - 0.424i)T \)
31 \( 1 + (0.860 + 0.509i)T \)
37 \( 1 + (0.187 - 0.982i)T \)
41 \( 1 + (-0.431 + 0.901i)T \)
43 \( 1 + (0.775 - 0.631i)T \)
47 \( 1 + (0.629 + 0.777i)T \)
53 \( 1 + (-0.928 + 0.370i)T \)
59 \( 1 + (0.853 - 0.520i)T \)
61 \( 1 + (-0.975 + 0.218i)T \)
67 \( 1 + (0.486 + 0.873i)T \)
71 \( 1 + (0.108 + 0.994i)T \)
73 \( 1 + (-0.567 + 0.823i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.0186 - 0.999i)T \)
89 \( 1 + (0.999 + 0.0292i)T \)
97 \( 1 + (-0.966 - 0.257i)T \)
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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

−18.30485538393044590975232438426, −17.303616172128307418498372781404, −16.308681232409880052609111573812, −15.42988448254495554299638377483, −14.89359661650331583783202469308, −14.31946014935122081982666019071, −13.68417885813713363539890677588, −13.19041480824190763962378302557, −12.20790173357274256985448533427, −11.879243765370471230743823341252, −10.9700380773004844371874757436, −10.64841866147564045825187652515, −9.62560538141453617779527141689, −8.71145428037583241875038983197, −8.28270405622899044601432223120, −7.68961498401183215137842564871, −6.7683157647293453883865340110, −5.93643150803758715016658408734, −4.85383211587144432498212665993, −4.22481701999185555690550151701, −3.61145205018977898319597227031, −2.872207382083829705381483665, −2.13445795681051815223059421693, −1.3126989008631263222296928879, −0.42850714908568181822477781580, 1.25661881794969149002630660868, 2.45087435635814732736579292834, 3.24583215794608344855520162498, 3.99855744291454440822089245865, 4.458337434315483257924036153440, 5.340949319380635365710533165988, 5.75186597751714250674762724532, 7.195577597651931497943629668888, 7.823973389196119050924435631530, 8.00751255017625116065088777554, 8.61684193347885089580747849127, 9.51271761215690615528224066118, 10.30108253032369460667570531462, 11.11145734307445019225092845273, 12.02778282860203020792766674757, 12.63105065102554121808611131468, 13.26695059975132728563562013294, 14.166581064469462280631126540357, 14.65670839935547620545660150056, 15.18319138506526390702787276659, 15.86456437335642425969975048967, 16.035722124900998210249916053444, 17.121351074803170847465289424833, 17.71738642613996966676415872846, 18.72677689866627295762381220797

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