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Properties

Label 1-4729-4729.1087-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.0921 - 0.995i$
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 + (0.975 − 0.221i)5-s + (−0.989 − 0.143i)6-s + (−0.908 + 0.417i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.589 + 0.808i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (−0.978 + 0.205i)13-s + (0.410 − 0.912i)14-s + (0.803 + 0.595i)15-s + (−0.963 − 0.267i)16-s + (−0.999 + 0.0318i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (0.975 − 0.221i)5-s + (−0.989 − 0.143i)6-s + (−0.908 + 0.417i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.589 + 0.808i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (−0.978 + 0.205i)13-s + (0.410 − 0.912i)14-s + (0.803 + 0.595i)15-s + (−0.963 − 0.267i)16-s + (−0.999 + 0.0318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0921 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0921 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.0921 - 0.995i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.0921 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3062119043 + 0.2791940753i\)
\(L(\frac12)\) \(\approx\) \(-0.3062119043 + 0.2791940753i\)
\(L(1)\) \(\approx\) \(0.5500461513 + 0.5016380510i\)
\(L(1)\) \(\approx\) \(0.5500461513 + 0.5016380510i\)

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 \( 1 + (0.975 - 0.221i)T \)
7 \( 1 + (-0.908 + 0.417i)T \)
11 \( 1 + (-0.698 + 0.715i)T \)
13 \( 1 + (-0.978 + 0.205i)T \)
17 \( 1 + (-0.999 + 0.0318i)T \)
19 \( 1 + (-0.244 + 0.969i)T \)
23 \( 1 + (-0.467 + 0.884i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.954 + 0.298i)T \)
37 \( 1 + (-0.119 - 0.992i)T \)
41 \( 1 + (-0.987 - 0.158i)T \)
43 \( 1 + (-0.166 + 0.986i)T \)
47 \( 1 + (-0.856 + 0.516i)T \)
53 \( 1 + (0.812 - 0.582i)T \)
59 \( 1 + (0.103 - 0.994i)T \)
61 \( 1 + (0.709 + 0.704i)T \)
67 \( 1 + (0.439 - 0.898i)T \)
71 \( 1 + (-0.732 + 0.681i)T \)
73 \( 1 + (0.763 + 0.645i)T \)
79 \( 1 - T \)
83 \( 1 + (0.967 - 0.252i)T \)
89 \( 1 + (0.998 + 0.0478i)T \)
97 \( 1 + (-0.908 + 0.417i)T \)
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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

−17.738926293252233595217286562234, −17.2481173924775847739679692518, −16.61596250844269094012075934514, −15.6633066859006864397078134933, −15.02184403377705805042806638776, −13.82991255521435823310737625775, −13.4471341581854711801538717523, −13.12435051496938101772322198165, −12.28660645430026045964989953547, −11.61904301394369160582256042983, −10.550771571138270234766153540899, −10.14391930852927450625301003477, −9.52853124058271726388446372829, −8.73218262219621528420666872017, −8.28278085788592386124142189648, −7.32639640842806701911844189170, −6.63590612142575799432795162202, −6.3385394783685000206231520420, −5.00411326816399553144163872951, −4.0065340353012248718148173873, −2.90087868713553594648340138380, −2.67974943334925335594223231537, −2.06194932026332229419651791259, −0.89185722320553482664388945567, −0.13813509064418751310416045044, 1.61104604530395893300753149251, 2.27366849887890734026433425026, 2.82764194310773087838330171730, 4.085861861578052562892242846644, 5.05097025270329214044158317311, 5.346098010719941103545057978774, 6.37401773323184195336670124315, 6.91764471680733279979280067555, 7.89062060547594122533213722776, 8.52588007843941079222305869200, 9.25385802362403029622416390671, 9.7903708910684475178329611766, 10.09435284710644864193405038455, 10.73775053041493555951893892027, 11.912117586360282135944233462523, 12.84653342433171755084510452448, 13.45588512581973542208137873601, 14.2126121375992682342889804584, 14.78561046542210573358716647232, 15.434440502221799030853652264684, 16.12314688711100967629513024396, 16.42899352852824293540974779869, 17.475533424900117534072160743294, 17.715267946089878294203971295, 18.67052392220257074774581779771

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