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Properties

Label 1-4729-4729.144-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.843 + 0.536i$
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  + (−0.880 + 0.474i)2-s + (−0.275 + 0.961i)3-s + (0.549 − 0.835i)4-s + (−0.944 − 0.328i)5-s + (−0.213 − 0.976i)6-s + (0.601 − 0.798i)7-s + (−0.0875 + 0.996i)8-s + (−0.848 − 0.529i)9-s + (0.987 − 0.158i)10-s + (0.366 + 0.930i)11-s + (0.651 + 0.758i)12-s + (−0.305 + 0.952i)13-s + (−0.150 + 0.988i)14-s + (0.576 − 0.817i)15-s + (−0.395 − 0.918i)16-s + (0.998 + 0.0478i)17-s + ⋯
L(s)  = 1  + (−0.880 + 0.474i)2-s + (−0.275 + 0.961i)3-s + (0.549 − 0.835i)4-s + (−0.944 − 0.328i)5-s + (−0.213 − 0.976i)6-s + (0.601 − 0.798i)7-s + (−0.0875 + 0.996i)8-s + (−0.848 − 0.529i)9-s + (0.987 − 0.158i)10-s + (0.366 + 0.930i)11-s + (0.651 + 0.758i)12-s + (−0.305 + 0.952i)13-s + (−0.150 + 0.988i)14-s + (0.576 − 0.817i)15-s + (−0.395 − 0.918i)16-s + (0.998 + 0.0478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.843 + 0.536i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.843 + 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9437517400 + 0.2748219601i\)
\(L(\frac12)\) \(\approx\) \(0.9437517400 + 0.2748219601i\)
\(L(1)\) \(\approx\) \(0.6333200793 + 0.2254068793i\)
\(L(1)\) \(\approx\) \(0.6333200793 + 0.2254068793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.880 + 0.474i)T \)
3 \( 1 + (-0.275 + 0.961i)T \)
5 \( 1 + (-0.944 - 0.328i)T \)
7 \( 1 + (0.601 - 0.798i)T \)
11 \( 1 + (0.366 + 0.930i)T \)
13 \( 1 + (-0.305 + 0.952i)T \)
17 \( 1 + (0.998 + 0.0478i)T \)
19 \( 1 + (0.915 + 0.402i)T \)
23 \( 1 + (0.0557 - 0.998i)T \)
29 \( 1 + (0.880 + 0.474i)T \)
31 \( 1 + (-0.439 - 0.898i)T \)
37 \( 1 + (0.821 - 0.569i)T \)
41 \( 1 + (0.971 - 0.236i)T \)
43 \( 1 + (0.509 - 0.860i)T \)
47 \( 1 + (0.687 + 0.726i)T \)
53 \( 1 + (-0.803 + 0.595i)T \)
59 \( 1 + (-0.589 + 0.808i)T \)
61 \( 1 + (0.921 + 0.388i)T \)
67 \( 1 + (0.103 - 0.994i)T \)
71 \( 1 + (0.901 - 0.431i)T \)
73 \( 1 + (0.495 - 0.868i)T \)
79 \( 1 - T \)
83 \( 1 + (0.927 + 0.373i)T \)
89 \( 1 + (-0.0717 - 0.997i)T \)
97 \( 1 + (0.601 - 0.798i)T \)
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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

−18.14802561614464685287822352081, −17.681277493039099021133440039622, −17.036919356502172670379651449124, −16.05398866846750855605995756154, −15.76580760153399225716942608120, −14.71528290201688739411483070991, −14.115967267970874248318765759471, −13.104776300738752092514155362192, −12.38619601011391796495602773769, −11.91513777278434979599763997497, −11.31002807943012761397707575463, −11.01264428916516818156711135540, −9.95795134917664571952082151983, −9.09743744103703168481988441873, −8.21635649446129096912033213533, −7.9762921046724271442216722187, −7.38135636860090207212323886253, −6.539599767583272058539319654109, −5.71748216411653356222141608444, −4.98143756676655132948073702337, −3.62045352833155864292625542227, −2.98851514481135776079311182489, −2.467258222367186917548517847541, −1.1796135809667414893854859746, −0.83528136786447378498421037535, 0.58511150394637324920820565160, 1.33603271868294535002285960748, 2.4969148257935726087155426085, 3.66508352071240222651818164879, 4.40516166077279391000803421838, 4.80577714096355592367237514750, 5.71134411509379052315875048264, 6.62807714608788355826026918175, 7.53255573061942923871534715951, 7.729633954708929975842552417799, 8.78843662221247832137358701931, 9.32945206143479933676938766247, 9.98253110063807107584814526416, 10.6941118618681758003293441660, 11.24268150454126890296106903323, 11.97755845072357399685863091133, 12.4206547433890644587665459785, 14.13132774183827266200713657950, 14.35427977900553782019021483612, 14.99473339053551362334095649049, 15.74822135001026558962355501332, 16.38589263222338711592581231550, 16.77546421334857657169931833598, 17.28379350078865058036284038926, 18.10991207481938924676326448705

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