L-function 1-4729-4729.1059-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	4729	$1$
good	2	$1 + (-0.576 - 0.817i)T$
	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	$1 + (-0.388 + 0.921i)T$

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