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Properties

Label 1-4729-4729.1710-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.0336 + 0.999i$
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  + (0.783 − 0.620i)2-s + (−0.984 − 0.174i)3-s + (0.229 − 0.973i)4-s + (0.809 − 0.586i)5-s + (−0.880 + 0.474i)6-s + (−0.818 + 0.573i)7-s + (−0.424 − 0.905i)8-s + (0.939 + 0.343i)9-s + (0.270 − 0.962i)10-s + (−0.954 − 0.298i)11-s + (−0.395 + 0.918i)12-s + (0.486 + 0.873i)13-s + (−0.285 + 0.958i)14-s + (−0.899 + 0.436i)15-s + (−0.894 − 0.446i)16-s + (0.690 + 0.722i)17-s + ⋯
L(s)  = 1  + (0.783 − 0.620i)2-s + (−0.984 − 0.174i)3-s + (0.229 − 0.973i)4-s + (0.809 − 0.586i)5-s + (−0.880 + 0.474i)6-s + (−0.818 + 0.573i)7-s + (−0.424 − 0.905i)8-s + (0.939 + 0.343i)9-s + (0.270 − 0.962i)10-s + (−0.954 − 0.298i)11-s + (−0.395 + 0.918i)12-s + (0.486 + 0.873i)13-s + (−0.285 + 0.958i)14-s + (−0.899 + 0.436i)15-s + (−0.894 − 0.446i)16-s + (0.690 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0336 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0336 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.0336 + 0.999i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1710, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.0336 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1529448786 + 0.1478893000i\)
\(L(\frac12)\) \(\approx\) \(0.1529448786 + 0.1478893000i\)
\(L(1)\) \(\approx\) \(0.9091466199 - 0.4522449907i\)
\(L(1)\) \(\approx\) \(0.9091466199 - 0.4522449907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.783 - 0.620i)T \)
3 \( 1 + (-0.984 - 0.174i)T \)
5 \( 1 + (0.809 - 0.586i)T \)
7 \( 1 + (-0.818 + 0.573i)T \)
11 \( 1 + (-0.954 - 0.298i)T \)
13 \( 1 + (0.486 + 0.873i)T \)
17 \( 1 + (0.690 + 0.722i)T \)
19 \( 1 + (-0.518 + 0.855i)T \)
23 \( 1 + (-0.275 - 0.961i)T \)
29 \( 1 + (0.783 + 0.620i)T \)
31 \( 1 + (0.763 + 0.645i)T \)
37 \( 1 + (0.593 + 0.804i)T \)
41 \( 1 + (-0.366 - 0.930i)T \)
43 \( 1 + (-0.545 - 0.838i)T \)
47 \( 1 + (-0.992 - 0.121i)T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (-0.869 - 0.493i)T \)
61 \( 1 + (-0.994 - 0.100i)T \)
67 \( 1 + (-0.999 - 0.00531i)T \)
71 \( 1 + (-0.375 - 0.926i)T \)
73 \( 1 + (-0.999 - 0.0265i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.983 + 0.179i)T \)
89 \( 1 + (-0.986 + 0.164i)T \)
97 \( 1 + (-0.818 + 0.573i)T \)
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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

−17.88515077996640954909047691886, −17.2726519974368179777667214575, −16.56042635846362400877806482139, −15.983515201553572671365772080473, −15.37077916295095942923605337305, −14.85000446940260042874471112805, −13.63419742910266537093441574351, −13.39436515159357757163081864495, −12.87609407634434942093975597902, −12.02820870536117358897934005593, −11.2382661508377135463223946164, −10.585422543948619753297652868444, −9.950295465095490986914447559506, −9.373954031086754205431865357, −8.01684373735660140813855506308, −7.38759520828762848462034499344, −6.77044274178363011568904580867, −5.96897398761558799147321709694, −5.75808591479569699408190419722, −4.815312786680866143155799766047, −4.20501354396543470175159682134, −2.97660159142382340019610784758, −2.85426701203360979979276050182, −1.36953385145832218562451311648, −0.04809439442874939697062029514, 1.192322964296749034807816605260, 1.81365452885779279268462071590, 2.637917686066022529310765075997, 3.57371664548502103422578704620, 4.49270777649960397102181972974, 5.14548345765292805949321024147, 5.76619253442971244947257622278, 6.36113180211180463797416008646, 6.690633547222583368775025945151, 8.16870854902358804338892545615, 8.89062686480046734253496419536, 9.90394665883589128611628645387, 10.29148864153481116023578285810, 10.777038629673809238908060164190, 11.96956711398452698841575899712, 12.2256550336111392490285622533, 12.82482914548756986824762831785, 13.43171338179236368887206167342, 13.9845035812384453483847429594, 14.95043567921220037072932756565, 15.75646053763402872259431153031, 16.48251676289216120466629515887, 16.621504896800209876340743362446, 17.83260216208081358972343521336, 18.49211665012719054095633509189

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