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Properties

Label 1-4729-4729.1103-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.999 + 0.0115i$
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  + (0.119 − 0.992i)2-s + (0.989 − 0.143i)3-s + (−0.971 − 0.236i)4-s + (−0.467 − 0.884i)5-s + (−0.0239 − 0.999i)6-s + (−0.0717 + 0.997i)7-s + (−0.351 + 0.936i)8-s + (0.959 − 0.283i)9-s + (−0.933 + 0.358i)10-s + (0.924 − 0.380i)11-s + (−0.995 − 0.0955i)12-s + (−0.848 − 0.529i)13-s + (0.981 + 0.190i)14-s + (−0.589 − 0.808i)15-s + (0.887 + 0.460i)16-s + (−0.868 + 0.495i)17-s + ⋯
L(s)  = 1  + (0.119 − 0.992i)2-s + (0.989 − 0.143i)3-s + (−0.971 − 0.236i)4-s + (−0.467 − 0.884i)5-s + (−0.0239 − 0.999i)6-s + (−0.0717 + 0.997i)7-s + (−0.351 + 0.936i)8-s + (0.959 − 0.283i)9-s + (−0.933 + 0.358i)10-s + (0.924 − 0.380i)11-s + (−0.995 − 0.0955i)12-s + (−0.848 − 0.529i)13-s + (0.981 + 0.190i)14-s + (−0.589 − 0.808i)15-s + (0.887 + 0.460i)16-s + (−0.868 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.999 + 0.0115i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.999 + 0.0115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009521044644 - 1.650416489i\)
\(L(\frac12)\) \(\approx\) \(0.009521044644 - 1.650416489i\)
\(L(1)\) \(\approx\) \(0.9104211220 - 0.8115228761i\)
\(L(1)\) \(\approx\) \(0.9104211220 - 0.8115228761i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.119 - 0.992i)T \)
3 \( 1 + (0.989 - 0.143i)T \)
5 \( 1 + (-0.467 - 0.884i)T \)
7 \( 1 + (-0.0717 + 0.997i)T \)
11 \( 1 + (0.924 - 0.380i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (-0.868 + 0.495i)T \)
19 \( 1 + (-0.954 + 0.298i)T \)
23 \( 1 + (0.941 - 0.336i)T \)
29 \( 1 + (0.992 - 0.119i)T \)
31 \( 1 + (-0.543 - 0.839i)T \)
37 \( 1 + (0.721 - 0.692i)T \)
41 \( 1 + (-0.852 + 0.522i)T \)
43 \( 1 + (-0.726 - 0.687i)T \)
47 \( 1 + (0.817 + 0.576i)T \)
53 \( 1 + (0.994 + 0.103i)T \)
59 \( 1 + (0.275 + 0.961i)T \)
61 \( 1 + (-0.608 - 0.793i)T \)
67 \( 1 + (0.651 - 0.758i)T \)
71 \( 1 + (0.921 - 0.388i)T \)
73 \( 1 + (0.395 - 0.918i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.536 + 0.843i)T \)
89 \( 1 + (-0.999 + 0.00797i)T \)
97 \( 1 + (0.0717 - 0.997i)T \)
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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

−18.54197732635535077569625888490, −17.6705588596442202603421539591, −17.08720589163564396375356680193, −16.42788636450757646180613063803, −15.59839472715678707468525119756, −15.053509924056032524298074964385, −14.55185552305195386686437612927, −14.03740008260723959829997617056, −13.46399151498713576765189037515, −12.72368450826864210160512659620, −11.83084246761737833263965716791, −10.89956024576524655013300111703, −10.08827738719010040875472915204, −9.56066543274114199335943837411, −8.75991478350810896295144849697, −8.17744276882518800159445483733, −7.20372720290720603418148521189, −6.84598326496332393576078651593, −6.67829165065260449497921135561, −5.04255084061991802886929487423, −4.36107688624320785725764795026, −3.94886009412166373749613596018, −3.14149299112837087950151340931, −2.30832609488876164101909521256, −1.08302523822515738912223219137, 0.39723156692690510034163787267, 1.43655643583034210091435468922, 2.17204249603230473335155856085, 2.79510443449088605714444671731, 3.68134957410388686338879088472, 4.30078252756499894903513781589, 4.935370123235119431129573698381, 5.867711661764209457500062178742, 6.77714334876475075258263516694, 7.988489734709765793014877712053, 8.41635268910332982311828883818, 9.08353843755301365033558871429, 9.34636935194258741373911386780, 10.31264158538266010648244517901, 11.15672578325118632317287160477, 11.99281241385316446356709596422, 12.46212955722239261441720791366, 12.93862650751619585655344351058, 13.57877653301314148945644061304, 14.48450158845178276295203680516, 15.13266579866962173365757257190, 15.37457836730405297037641217140, 16.671909776092077796310170381472, 17.16461631443164184934695262402, 18.11716522834879762575091307613

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