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Properties

Label 1-4729-4729.1059-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.997 + 0.0634i$
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  + (−0.576 − 0.817i)2-s + (−0.410 + 0.912i)3-s + (−0.336 + 0.941i)4-s + (0.224 + 0.974i)5-s + (0.981 − 0.190i)6-s + (0.890 − 0.455i)7-s + (0.963 − 0.267i)8-s + (−0.663 − 0.748i)9-s + (0.667 − 0.744i)10-s + (0.0159 − 0.999i)11-s + (−0.721 − 0.692i)12-s + (0.962 + 0.272i)13-s + (−0.885 − 0.465i)14-s + (−0.980 − 0.195i)15-s + (−0.773 − 0.633i)16-s + (−0.886 + 0.462i)17-s + ⋯
L(s)  = 1  + (−0.576 − 0.817i)2-s + (−0.410 + 0.912i)3-s + (−0.336 + 0.941i)4-s + (0.224 + 0.974i)5-s + (0.981 − 0.190i)6-s + (0.890 − 0.455i)7-s + (0.963 − 0.267i)8-s + (−0.663 − 0.748i)9-s + (0.667 − 0.744i)10-s + (0.0159 − 0.999i)11-s + (−0.721 − 0.692i)12-s + (0.962 + 0.272i)13-s + (−0.885 − 0.465i)14-s + (−0.980 − 0.195i)15-s + (−0.773 − 0.633i)16-s + (−0.886 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.997 + 0.0634i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.997 + 0.0634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006411155938 + 0.2018800165i\)
\(L(\frac12)\) \(\approx\) \(0.006411155938 + 0.2018800165i\)
\(L(1)\) \(\approx\) \(0.6615779235 + 0.04933079684i\)
\(L(1)\) \(\approx\) \(0.6615779235 + 0.04933079684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.576 - 0.817i)T \)
3 \( 1 + (-0.410 + 0.912i)T \)
5 \( 1 + (0.224 + 0.974i)T \)
7 \( 1 + (0.890 - 0.455i)T \)
11 \( 1 + (0.0159 - 0.999i)T \)
13 \( 1 + (0.962 + 0.272i)T \)
17 \( 1 + (-0.886 + 0.462i)T \)
19 \( 1 + (0.946 - 0.323i)T \)
23 \( 1 + (-0.388 + 0.921i)T \)
29 \( 1 + (-0.817 - 0.576i)T \)
31 \( 1 + (-0.992 - 0.119i)T \)
37 \( 1 + (-0.631 + 0.775i)T \)
41 \( 1 + (0.952 + 0.305i)T \)
43 \( 1 + (-0.733 + 0.679i)T \)
47 \( 1 + (0.661 - 0.749i)T \)
53 \( 1 + (-0.953 - 0.300i)T \)
59 \( 1 + (0.375 - 0.926i)T \)
61 \( 1 + (-0.863 - 0.504i)T \)
67 \( 1 + (-0.904 - 0.427i)T \)
71 \( 1 + (-0.540 - 0.841i)T \)
73 \( 1 + (-0.593 + 0.804i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.760 + 0.649i)T \)
89 \( 1 + (0.896 + 0.443i)T \)
97 \( 1 + (-0.890 + 0.455i)T \)
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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

−17.750660088199961239660576446568, −17.45898125441341200602651047262, −16.4473155521156706985271497826, −16.10462919422756552742958185038, −15.28576865495916611289038061279, −14.445815181239626761293249811350, −13.87125684934388080005042424843, −13.16574734176638459038275027327, −12.49290699337650877477653767836, −11.828958311541740269974831289560, −10.99290294859696741726616137794, −10.41350091468903661292641768507, −9.14067692870193097437962498575, −8.96913687823880049002087165295, −8.12909606616742304054559133641, −7.520207157469348811489310402805, −6.95969376159318243326804956125, −5.87606758511867212637720239266, −5.58446232699753599359454296424, −4.83809491091233158211412492509, −4.16895965753639673083856413591, −2.44457998018281230869769779574, −1.63407103383765925791977564863, −1.281676813321053744144041346524, −0.07246350823248813231066890664, 1.21775373619711513669681449930, 1.98636750906086952206343486049, 3.10663164061825712638960606743, 3.615115452912644880571358732386, 4.17944066407671673354644868031, 5.17027852266800917185090251831, 5.96675953968982635496640450441, 6.77768383308494481640915505401, 7.72019928116119189558619294304, 8.354338128002532206729595787809, 9.210249956558836502368305162643, 9.67680540669442853687726985613, 10.70857547367024597321443605424, 10.90506964416112375522969865515, 11.41095851295896254855867351204, 11.88784705465524747971819429143, 13.35118883065751341263936295853, 13.643672187184260838377035595513, 14.40508986719103752707216382899, 15.23747588733612083361819428615, 15.95187418184040883390919812367, 16.600078559409313814483565720588, 17.36978047414484279728505532107, 17.83142718318361695431611574592, 18.36624784941327892606473354328

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