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Properties

Label 1-4729-4729.1024-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.155 + 0.987i$
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  + (0.742 − 0.669i)2-s + (−0.536 + 0.843i)3-s + (0.103 − 0.994i)4-s + (0.260 + 0.965i)5-s + (0.166 + 0.986i)6-s + (−0.481 + 0.876i)7-s + (−0.589 − 0.808i)8-s + (−0.424 − 0.905i)9-s + (0.839 + 0.543i)10-s + (0.395 − 0.918i)11-s + (0.783 + 0.620i)12-s + (0.721 + 0.692i)13-s + (0.229 + 0.973i)14-s + (−0.954 − 0.298i)15-s + (−0.978 − 0.205i)16-s + (−0.467 − 0.884i)17-s + ⋯
L(s)  = 1  + (0.742 − 0.669i)2-s + (−0.536 + 0.843i)3-s + (0.103 − 0.994i)4-s + (0.260 + 0.965i)5-s + (0.166 + 0.986i)6-s + (−0.481 + 0.876i)7-s + (−0.589 − 0.808i)8-s + (−0.424 − 0.905i)9-s + (0.839 + 0.543i)10-s + (0.395 − 0.918i)11-s + (0.783 + 0.620i)12-s + (0.721 + 0.692i)13-s + (0.229 + 0.973i)14-s + (−0.954 − 0.298i)15-s + (−0.978 − 0.205i)16-s + (−0.467 − 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.155 + 0.987i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.155 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9879265880 + 1.155604161i\)
\(L(\frac12)\) \(\approx\) \(0.9879265880 + 1.155604161i\)
\(L(1)\) \(\approx\) \(1.198016889 + 0.1364746853i\)
\(L(1)\) \(\approx\) \(1.198016889 + 0.1364746853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.742 - 0.669i)T \)
3 \( 1 + (-0.536 + 0.843i)T \)
5 \( 1 + (0.260 + 0.965i)T \)
7 \( 1 + (-0.481 + 0.876i)T \)
11 \( 1 + (0.395 - 0.918i)T \)
13 \( 1 + (0.721 + 0.692i)T \)
17 \( 1 + (-0.467 - 0.884i)T \)
19 \( 1 + (0.522 + 0.852i)T \)
23 \( 1 + (-0.675 + 0.737i)T \)
29 \( 1 + (-0.742 - 0.669i)T \)
31 \( 1 + (0.639 - 0.768i)T \)
37 \( 1 + (0.601 + 0.798i)T \)
41 \( 1 + (-0.651 - 0.758i)T \)
43 \( 1 + (0.830 + 0.556i)T \)
47 \( 1 + (-0.915 + 0.402i)T \)
53 \( 1 + (0.663 - 0.748i)T \)
59 \( 1 + (0.927 - 0.373i)T \)
61 \( 1 + (-0.135 + 0.990i)T \)
67 \( 1 + (0.967 + 0.252i)T \)
71 \( 1 + (0.939 + 0.343i)T \)
73 \( 1 + (0.290 + 0.956i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.732 + 0.681i)T \)
89 \( 1 + (0.0557 + 0.998i)T \)
97 \( 1 + (-0.481 + 0.876i)T \)
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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

−17.82415032356947407621585056285, −17.15312402848163901706061102653, −16.73248067404582624036120697989, −16.03942151102122063705719657564, −15.45653147506431848740831651981, −14.43404879134878307854147091516, −13.76526026616939831963793624000, −13.17916498983082380474323540900, −12.73365717315668929357371926351, −12.3298263973465351425934352490, −11.417710636044912427076759245322, −10.69336682320929164932208883421, −9.78458753485485836078305818998, −8.77078097307185108691207656959, −8.22405720077897176240347242160, −7.428237007759190543786709312731, −6.82528008255962094182265361324, −6.20623605537617627944889127146, −5.56171411206705073022598194656, −4.739574409171289199090581502389, −4.20855568821587611051749225023, −3.2969635359351352626544940035, −2.19326887903128040905767295684, −1.38273868943725560337507979591, −0.35971552292813593834021973835, 1.0528418593083362926062018697, 2.22172230445010476122423672336, 2.8703701039683762314805578202, 3.73177314705186214533346040171, 3.96165541996457443664282756676, 5.25557410634408633384857036753, 5.75908196519309636894122287927, 6.29536942047793655657181458851, 6.81665847314304782383243848022, 8.2378781520599507355359979733, 9.28933081954541836550878632651, 9.62567080285650190913395435018, 10.22781345518758804357527223943, 11.22262015569239724478197565730, 11.57051727935628589515506978563, 11.81829032868170038775121284976, 13.0359043202943219666610871842, 13.72694780019028649284286945975, 14.27649324763257493419255265563, 14.932544241261310890416894175972, 15.74535649234120665767549299577, 15.96365381656971800091530513563, 16.86215867850101973915451208122, 17.91928295720859482364320113246, 18.506591416196819483832219677528

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