↑

Properties

Label 1-847-847.381-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.538 - 0.842i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.935 − 0.353i)2-s + (0.104 − 0.994i)3-s + (0.749 − 0.662i)4-s + (0.830 − 0.556i)5-s + (−0.254 − 0.967i)6-s + (0.466 − 0.884i)8-s + (−0.978 − 0.207i)9-s + (0.580 − 0.814i)10-s + (−0.580 − 0.814i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)15-s + (0.123 − 0.992i)16-s + (0.997 + 0.0760i)17-s + (−0.988 + 0.151i)18-s + (−0.761 + 0.647i)19-s + (0.254 − 0.967i)20-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + (0.104 − 0.994i)3-s + (0.749 − 0.662i)4-s + (0.830 − 0.556i)5-s + (−0.254 − 0.967i)6-s + (0.466 − 0.884i)8-s + (−0.978 − 0.207i)9-s + (0.580 − 0.814i)10-s + (−0.580 − 0.814i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)15-s + (0.123 − 0.992i)16-s + (0.997 + 0.0760i)17-s + (−0.988 + 0.151i)18-s + (−0.761 + 0.647i)19-s + (0.254 − 0.967i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.538 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574757528 - 2.876681575i\)
\(L(\frac12)\) \(\approx\) \(1.574757528 - 2.876681575i\)
\(L(1)\) \(\approx\) \(1.687318422 - 1.387192823i\)
\(L(1)\) \(\approx\) \(1.687318422 - 1.387192823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.935 - 0.353i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (0.830 - 0.556i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (0.997 + 0.0760i)T \)
19 \( 1 + (-0.761 + 0.647i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (-0.610 + 0.791i)T \)
31 \( 1 + (-0.953 - 0.299i)T \)
37 \( 1 + (0.861 - 0.508i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.988 - 0.151i)T \)
53 \( 1 + (0.123 + 0.992i)T \)
59 \( 1 + (-0.548 - 0.836i)T \)
61 \( 1 + (0.161 + 0.986i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (-0.290 - 0.956i)T \)
79 \( 1 + (-0.345 + 0.938i)T \)
83 \( 1 + (0.516 + 0.856i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
97 \( 1 + (-0.897 + 0.441i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.28027584382246782148815455606, −21.626642851908923750860401277567, −21.00228985383295974304897559109, −20.516913677828391926684238242027, −19.35938962485337644076455240622, −18.272885169777197877034031675524, −17.17103895426215253002931362568, −16.735175257331045828253509997419, −15.75962907254637451831386270451, −14.97110209758402934193150320618, −14.51979299635081748144727067333, −13.54113469922303984704477307257, −13.03565654965398770679738803276, −11.64111677456740224935464229474, −10.97245207246256077763511225788, −10.25945699348214115944397880258, −9.21618045993466729953783118204, −8.31818893342523898775312452016, −7.15381841212957287466044656515, −6.15554947063219393676883394120, −5.53585532916916355854606117884, −4.66850089168998248229294868493, −3.54673878756701377539899223112, −2.990603508235757289461054615990, −1.83201731890839002550743004498, 1.16698558350318152810918120369, 1.71840960826857660990064812057, 2.81562280840003876107342275835, 3.791693494884896266137308765506, 5.092505509852133449944724370987, 5.86464865928073193173897982937, 6.43962448639790788956021804388, 7.49955820894113751565384193892, 8.57010702188274486870475584231, 9.494754640048838371978063826405, 10.6268383282768850732847099225, 11.3880884896526333714106679537, 12.487284542407725915923865126945, 12.86762884448580070330040124271, 13.56002119117283218065272236946, 14.367093840301051350496590701525, 14.96678545382277958473996259147, 16.46836418120945317526356215310, 16.791715202491468747802589361398, 18.13219082011010087308457494748, 18.67655311749055980511672356830, 19.56250024755970180512364136973, 20.479728091989163074606927816560, 20.9590314537442091035893259792, 21.745517798288358887035959458863

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