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Properties

Label 1-4729-4729.525-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.104 + 0.994i$
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  + (0.839 − 0.543i)2-s + (−0.366 + 0.930i)3-s + (0.410 − 0.912i)4-s + (0.135 − 0.990i)5-s + (0.198 + 0.980i)6-s + (−0.563 + 0.826i)7-s + (−0.150 − 0.988i)8-s + (−0.732 − 0.681i)9-s + (−0.424 − 0.905i)10-s + (−0.803 + 0.595i)11-s + (0.698 + 0.715i)12-s + (−0.894 − 0.446i)13-s + (−0.0239 + 0.999i)14-s + (0.872 + 0.488i)15-s + (−0.663 − 0.748i)16-s + (0.793 − 0.608i)17-s + ⋯
L(s)  = 1  + (0.839 − 0.543i)2-s + (−0.366 + 0.930i)3-s + (0.410 − 0.912i)4-s + (0.135 − 0.990i)5-s + (0.198 + 0.980i)6-s + (−0.563 + 0.826i)7-s + (−0.150 − 0.988i)8-s + (−0.732 − 0.681i)9-s + (−0.424 − 0.905i)10-s + (−0.803 + 0.595i)11-s + (0.698 + 0.715i)12-s + (−0.894 − 0.446i)13-s + (−0.0239 + 0.999i)14-s + (0.872 + 0.488i)15-s + (−0.663 − 0.748i)16-s + (0.793 − 0.608i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.104 + 0.994i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.104 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3085166042 + 0.3426644837i\)
\(L(\frac12)\) \(\approx\) \(0.3085166042 + 0.3426644837i\)
\(L(1)\) \(\approx\) \(1.045104403 - 0.2361526013i\)
\(L(1)\) \(\approx\) \(1.045104403 - 0.2361526013i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.839 - 0.543i)T \)
3 \( 1 + (-0.366 + 0.930i)T \)
5 \( 1 + (0.135 - 0.990i)T \)
7 \( 1 + (-0.563 + 0.826i)T \)
11 \( 1 + (-0.803 + 0.595i)T \)
13 \( 1 + (-0.894 - 0.446i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (-0.908 + 0.417i)T \)
23 \( 1 + (0.971 - 0.236i)T \)
29 \( 1 + (-0.839 - 0.543i)T \)
31 \( 1 + (-0.0717 - 0.997i)T \)
37 \( 1 + (0.995 - 0.0955i)T \)
41 \( 1 + (-0.991 - 0.127i)T \)
43 \( 1 + (0.880 - 0.474i)T \)
47 \( 1 + (0.119 + 0.992i)T \)
53 \( 1 + (0.182 + 0.983i)T \)
59 \( 1 + (0.229 + 0.973i)T \)
61 \( 1 + (0.305 - 0.952i)T \)
67 \( 1 + (0.627 - 0.778i)T \)
71 \( 1 + (-0.336 + 0.941i)T \)
73 \( 1 + (-0.244 + 0.969i)T \)
79 \( 1 - T \)
83 \( 1 + (0.495 + 0.868i)T \)
89 \( 1 + (0.830 - 0.556i)T \)
97 \( 1 + (-0.563 + 0.826i)T \)
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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

−17.77316361238333599170621613808, −17.25016127705294155141617393831, −16.62509151131874899198350160095, −16.13954384517416683988382884383, −14.95493150355794150146231894374, −14.64561844232258844867924681203, −13.890783974704854220261331969408, −13.20365960890071271716851665327, −12.9239166430712128706091576479, −12.08043606764807422558126709291, −11.24823912055006940353222880769, −10.777348802808259649688533675416, −10.079103496513130722566100771698, −8.85298335956773813956016863758, −7.9383588835948288023248091767, −7.37894257963316368844084495714, −6.85680896825790672616922366422, −6.36751261761397823523762803469, −5.57076359643802895030554108266, −4.931546366941194825614093126349, −3.82843738210337239085548530132, −3.08957581353224570459626233723, −2.54371198074249658025384056425, −1.59998889545964825906333697706, −0.10259112082996096340675255582, 0.92048379095993799905612807189, 2.32245100779649426148985079245, 2.65701867687599620240000350347, 3.69821628083867734634509965964, 4.45708989586349999217751756921, 5.049958697950459514473499227913, 5.606823543605713644283064441591, 6.03730829988489836791896535291, 7.20475199402876368486204126972, 8.168886022854858920185745498481, 9.225720592402058677037194101546, 9.60922073437852588575569694699, 10.13410168751089719345975815958, 10.93994545745495920294507681286, 11.76104656991018069108051595194, 12.38312736948327324362081764036, 12.7398268978858378557876373959, 13.40263527379066603311433223066, 14.512104939988874369192429655043, 15.06390437363371245069971540430, 15.5107305935498604716134731331, 16.19418336045157026343065639007, 16.85211934049811901525069948702, 17.447152499141063033431655842306, 18.613497242218521336148491027041

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