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Properties

Label 1-4729-4729.1072-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.892 - 0.450i$
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  + (0.549 + 0.835i)2-s + (−0.848 + 0.529i)3-s + (−0.395 + 0.918i)4-s + (−0.929 − 0.368i)5-s + (−0.908 − 0.417i)6-s + (0.970 − 0.242i)7-s + (−0.984 + 0.174i)8-s + (0.439 − 0.898i)9-s + (−0.203 − 0.979i)10-s + (−0.732 − 0.681i)11-s + (−0.150 − 0.988i)12-s + (0.910 + 0.412i)13-s + (0.735 + 0.677i)14-s + (0.983 − 0.179i)15-s + (−0.687 − 0.726i)16-s + (−0.580 − 0.814i)17-s + ⋯
L(s)  = 1  + (0.549 + 0.835i)2-s + (−0.848 + 0.529i)3-s + (−0.395 + 0.918i)4-s + (−0.929 − 0.368i)5-s + (−0.908 − 0.417i)6-s + (0.970 − 0.242i)7-s + (−0.984 + 0.174i)8-s + (0.439 − 0.898i)9-s + (−0.203 − 0.979i)10-s + (−0.732 − 0.681i)11-s + (−0.150 − 0.988i)12-s + (0.910 + 0.412i)13-s + (0.735 + 0.677i)14-s + (0.983 − 0.179i)15-s + (−0.687 − 0.726i)16-s + (−0.580 − 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.892 - 0.450i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1072, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.892 - 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05054645999 + 0.2121209901i\)
\(L(\frac12)\) \(\approx\) \(-0.05054645999 + 0.2121209901i\)
\(L(1)\) \(\approx\) \(0.6785750088 + 0.3425712715i\)
\(L(1)\) \(\approx\) \(0.6785750088 + 0.3425712715i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.549 + 0.835i)T \)
3 \( 1 + (-0.848 + 0.529i)T \)
5 \( 1 + (-0.929 - 0.368i)T \)
7 \( 1 + (0.970 - 0.242i)T \)
11 \( 1 + (-0.732 - 0.681i)T \)
13 \( 1 + (0.910 + 0.412i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.976 - 0.216i)T \)
23 \( 1 + (-0.993 + 0.111i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (-0.614 - 0.788i)T \)
37 \( 1 + (0.635 - 0.772i)T \)
41 \( 1 + (0.887 + 0.460i)T \)
43 \( 1 + (-0.518 - 0.855i)T \)
47 \( 1 + (-0.836 + 0.547i)T \)
53 \( 1 + (-0.973 - 0.226i)T \)
59 \( 1 + (-0.671 - 0.741i)T \)
61 \( 1 + (0.270 + 0.962i)T \)
67 \( 1 + (0.311 - 0.950i)T \)
71 \( 1 + (-0.988 + 0.153i)T \)
73 \( 1 + (0.999 + 0.0106i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (0.370 + 0.928i)T \)
97 \( 1 + (0.970 - 0.242i)T \)
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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

−17.89641129906033280202359363929, −17.55453126969300906073972068193, −16.24804397943148590708985552917, −15.69649314110357438574323683950, −14.96198177404018597758121321978, −14.46765166663934215103694054001, −13.527410113901131435258498952476, −12.70792687833298312016014338815, −12.48085166053699982226423727203, −11.6360521591984539418374875342, −11.08142108633184926683525029653, −10.62936238937338421404334323367, −10.14314673407378062531451592878, −8.670477856973532713230956151112, −8.2066266639162224940975658136, −7.44525213468856695302152779552, −6.43091386951914529421320126464, −5.979353393877707080785176385180, −4.93518039322126259348200759445, −4.56370045350550421958991225308, −3.791550199482625036244890105602, −2.77447297021604321078598076435, −1.89281056435716963105766795959, −1.35620148153896716392152298135, −0.07172914577905807072754697290, 0.8147586045901983261689234523, 2.27938568041196310141586003377, 3.49133713513401974931690245995, 4.16112758541275194797688914619, 4.55176043823499779335580834397, 5.25497729635010154915628644900, 6.01212038667678760621053758593, 6.63947331539294106276568754271, 7.61142271758995966902576932321, 8.10423158537791234278313572836, 8.76842634141332877173065669732, 9.531500865111615769641493881433, 10.79493590146302706744881893170, 11.259149387844140082919186467070, 11.655726325372628095901578478346, 12.521841875237785312254861350544, 13.20078330870481334437657928221, 13.92177413818144115775090151504, 14.69882622320389843759938876436, 15.468900315722037258809107349037, 15.81890365539979088847300999111, 16.43298999926958203377758626471, 16.91062092146730745923676428573, 17.82341311043214391373195444381, 18.18471900961693882726928372043

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