↑

Properties

Label 1-436-436.63-r1-0-0
Degree $1$
Conductor $436$
Sign $0.0319 + 0.999i$
Analytic cond. $46.8547$
Root an. cond. $46.8547$
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  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + 17-s − 19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + 17-s − 19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 436 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0319 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 436 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0319 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(436\)    =    \(2^{2} \cdot 109\)
Sign: $0.0319 + 0.999i$
Analytic conductor: \(46.8547\)
Root analytic conductor: \(46.8547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{436} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 436,\ (1:\ ),\ 0.0319 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8598907496 + 0.8328248290i\)
\(L(\frac12)\) \(\approx\) \(0.8598907496 + 0.8328248290i\)
\(L(1)\) \(\approx\) \(1.041849951 - 0.03332614146i\)
\(L(1)\) \(\approx\) \(1.041849951 - 0.03332614146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
109 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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

−23.8191600649126198225294702978, −22.62601586455645598721072722093, −21.79776040243411424569747938332, −21.06698042474065454722801452363, −20.38211348163417602860254260677, −19.435897296127701163917546542708, −18.79569595787510900395286368214, −17.2904023341216856191821313977, −16.67471667215805704329173121211, −15.695571282928460086871785226273, −15.07519387662231596059969663672, −14.2409759765562815749995032874, −13.10879344253515758295997371266, −12.04126281723619757591909576392, −11.34564040816403676553567759229, −10.11977008390213369055620459158, −9.25139527181447380042548951621, −8.20780295992812486144477447873, −8.02904904818210266574682656178, −5.949166608232078997862291095, −5.18604144828253191645795877834, −4.19746298721578512531573351377, −3.22844048435266511357811680833, −1.96188113420377354013363287829, −0.29240788380187964690197959242, 1.34138178347941722327927095221, 2.32096160641042425654643928324, 3.61145783885322400857605099087, 4.438697400809339889637345078, 6.17868006273480157565932886279, 7.1776760744957467482864098606, 7.48697506542423838172057209619, 8.62481596428560925226637629103, 9.84223207023826826789033581513, 10.8078093457911810702408781449, 11.89964780140489099349059660735, 12.467700880394132959178580599525, 13.84051831912381887396437242156, 14.452252154316645056504385864325, 14.907053574226534992881846673511, 16.363409909766623720895151776454, 17.41495368765748319257899033208, 18.05678392816226295729554040632, 19.11193550541580947497737689792, 19.598588174860566174189607687604, 20.4536690317802448054034621221, 21.45028391875852678032522474922, 22.63244204326866634306950679871, 23.4719264771348195321732124383, 23.84315769035180537784846458313

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