Properties

Label 1-4729-4729.1026-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: $\chi_{4729} (1026, \cdot )$
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

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

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