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Properties

Label 1-847-847.325-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.820 − 0.572i)2-s + (−0.669 + 0.743i)3-s + (0.345 + 0.938i)4-s + (−0.548 + 0.836i)5-s + (0.974 − 0.226i)6-s + (0.254 − 0.967i)8-s + (−0.104 − 0.994i)9-s + (0.928 − 0.371i)10-s + (−0.928 − 0.371i)12-s + (0.696 + 0.717i)13-s + (−0.254 − 0.967i)15-s + (−0.761 + 0.647i)16-s + (0.861 + 0.508i)17-s + (−0.483 + 0.875i)18-s + (−0.217 − 0.976i)19-s + (−0.974 − 0.226i)20-s + ⋯
L(s)  = 1  + (−0.820 − 0.572i)2-s + (−0.669 + 0.743i)3-s + (0.345 + 0.938i)4-s + (−0.548 + 0.836i)5-s + (0.974 − 0.226i)6-s + (0.254 − 0.967i)8-s + (−0.104 − 0.994i)9-s + (0.928 − 0.371i)10-s + (−0.928 − 0.371i)12-s + (0.696 + 0.717i)13-s + (−0.254 − 0.967i)15-s + (−0.761 + 0.647i)16-s + (0.861 + 0.508i)17-s + (−0.483 + 0.875i)18-s + (−0.217 − 0.976i)19-s + (−0.974 − 0.226i)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: $\chi_{847} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.538 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3919407631 - 0.2147968724i\)
\(L(\frac12)\) \(\approx\) \(0.3919407631 - 0.2147968724i\)
\(L(1)\) \(\approx\) \(0.5124249332 + 0.01341365986i\)
\(L(1)\) \(\approx\) \(0.5124249332 + 0.01341365986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.820 - 0.572i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.548 + 0.836i)T \)
13 \( 1 + (0.696 + 0.717i)T \)
17 \( 1 + (0.861 + 0.508i)T \)
19 \( 1 + (-0.217 - 0.976i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (0.532 - 0.846i)T \)
37 \( 1 + (-0.830 + 0.556i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.483 - 0.875i)T \)
53 \( 1 + (-0.761 - 0.647i)T \)
59 \( 1 + (-0.797 - 0.603i)T \)
61 \( 1 + (-0.905 - 0.424i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.879 - 0.475i)T \)
79 \( 1 + (0.625 + 0.780i)T \)
83 \( 1 + (0.610 + 0.791i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (0.998 - 0.0570i)T \)
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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

−22.81684101934054202863245326534, −21.23804070212553028366280140022, −20.39971830151233954832474934046, −19.61585780366910952088082250141, −18.860543080911030217452797914145, −18.25095006460668526248064351618, −17.343338575657712547344574267423, −16.74409305537682158543471217754, −16.05739426470505718763191892810, −15.35822013247416687693519925890, −14.17781044107979102167765125733, −13.27788459031456630317862925564, −12.35698824202749082771891338058, −11.606746435603471334780063432, −10.79571421334488788291297268601, −9.84912012161211499560552043491, −8.786834093586773605057878503804, −7.91372704621533690794605631466, −7.53899964568873563849782009156, −6.360201095489024946243591989569, −5.54452915570593806119521567595, −4.91865440299570659591590436292, −3.356834401933723900642351898466, −1.65180434437810721695535207386, −1.02533472546994802615661876486, 0.359688163476353493718085808479, 1.895831102404448231244674893656, 3.24740468512861159268470446602, 3.782084316197739130822200312283, 4.84500015587202101627248650569, 6.35263826282576766713139626916, 6.856234616119063639994002169552, 8.05901733287371765769816443555, 8.90379414571670573872962781897, 9.862496384806253363380875202877, 10.57162266663058165812022904967, 11.23304612223734758952070059324, 11.78436098769548446860170448100, 12.681895402268218487142262049093, 13.9236621953972915776696530351, 15.14355701423889267986544318678, 15.58084081979942380479241004231, 16.71377579922038690441595566436, 17.00353472822735291518651954026, 18.239737000054687823682285470109, 18.6424729195273903611385872650, 19.475486814553520854235818037504, 20.49395928151204293354855361934, 21.11459885589451323350063969813, 21.95487213429833607176568807052

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