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Properties

Label 1-4729-4729.18-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.631 - 0.775i$
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  + (−0.0398 + 0.999i)2-s + (0.998 − 0.0478i)3-s + (−0.996 − 0.0796i)4-s + (0.773 − 0.633i)5-s + (0.00797 + 0.999i)6-s + (0.0239 − 0.999i)7-s + (0.119 − 0.992i)8-s + (0.995 − 0.0955i)9-s + (0.601 + 0.798i)10-s + (−0.608 − 0.793i)11-s + (−0.999 − 0.0318i)12-s + (0.651 − 0.758i)13-s + (0.997 + 0.0637i)14-s + (0.742 − 0.669i)15-s + (0.987 + 0.158i)16-s + (0.343 − 0.939i)17-s + ⋯
L(s)  = 1  + (−0.0398 + 0.999i)2-s + (0.998 − 0.0478i)3-s + (−0.996 − 0.0796i)4-s + (0.773 − 0.633i)5-s + (0.00797 + 0.999i)6-s + (0.0239 − 0.999i)7-s + (0.119 − 0.992i)8-s + (0.995 − 0.0955i)9-s + (0.601 + 0.798i)10-s + (−0.608 − 0.793i)11-s + (−0.999 − 0.0318i)12-s + (0.651 − 0.758i)13-s + (0.997 + 0.0637i)14-s + (0.742 − 0.669i)15-s + (0.987 + 0.158i)16-s + (0.343 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.695993542 - 1.281350506i\)
\(L(\frac12)\) \(\approx\) \(2.695993542 - 1.281350506i\)
\(L(1)\) \(\approx\) \(1.627793380 + 0.02716245005i\)
\(L(1)\) \(\approx\) \(1.627793380 + 0.02716245005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.0398 + 0.999i)T \)
3 \( 1 + (0.998 - 0.0478i)T \)
5 \( 1 + (0.773 - 0.633i)T \)
7 \( 1 + (0.0239 - 0.999i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (0.651 - 0.758i)T \)
17 \( 1 + (0.343 - 0.939i)T \)
19 \( 1 + (0.410 - 0.912i)T \)
23 \( 1 + (-0.595 - 0.803i)T \)
29 \( 1 + (0.999 - 0.0398i)T \)
31 \( 1 + (0.190 + 0.981i)T \)
37 \( 1 + (0.967 - 0.252i)T \)
41 \( 1 + (-0.983 + 0.182i)T \)
43 \( 1 + (0.267 + 0.963i)T \)
47 \( 1 + (-0.313 - 0.949i)T \)
53 \( 1 + (-0.529 + 0.848i)T \)
59 \( 1 + (0.908 + 0.417i)T \)
61 \( 1 + (0.737 - 0.675i)T \)
67 \( 1 + (0.959 - 0.283i)T \)
71 \( 1 + (-0.380 + 0.924i)T \)
73 \( 1 + (-0.135 + 0.990i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.944 + 0.328i)T \)
89 \( 1 + (0.502 + 0.864i)T \)
97 \( 1 + (-0.0239 + 0.999i)T \)
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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

−18.45562585368086775647677258798, −17.94986881173773496136308099894, −17.232082703248233026516119039010, −16.10497076949611993761449774342, −15.36500852158115687724500032798, −14.60310672510631876258881170388, −14.284560381260623500807706190028, −13.3950555915378942172597412940, −13.0252238747496251530747595543, −12.18167584334759818192315719659, −11.57234585797734664282165615455, −10.60045526191482635049800168032, −9.88221305541247110933995994193, −9.70618405517776819417137399357, −8.81442468093176607094361420825, −8.19475218060830716967369517545, −7.53574255854447346312274663355, −6.36964553900871685846039380443, −5.69714082571176860335817216573, −4.81774530228440080061535535640, −3.90548549864674647783437316471, −3.27287110082795017071351983315, −2.50284530592352731780767056745, −1.88258625283482379052492002844, −1.50167304467959926343725210284, 0.786079584447999149658000722296, 1.09864513962271461391556104907, 2.59213999916865314491046865248, 3.24204376707532107813116128139, 4.173687163416561358961627129641, 4.88507382847574516556166325633, 5.49233614922703518805791241136, 6.50312855390697161344649373540, 7.00060528630293861229443361484, 8.02877780099369587543386719707, 8.27295368834060628134175845746, 8.96304443488370580039336082826, 9.89743335535902344708619620735, 10.10468365699946523594198846060, 11.087845368360953051563454668129, 12.47247176760260410361855531246, 13.07798753232736708533605496804, 13.62427035085373406310398580223, 13.94567358178992125211577556093, 14.52775050714634252235812478097, 15.56799607437183036880110830998, 16.13379451202347727829094360063, 16.42084906954018186975214366057, 17.440847968429851015655566863241, 18.03612868568880658795596919084

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