↑

Properties

Label 1-4235-4235.522-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.549 + 0.835i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.572 + 0.820i)2-s + (−0.743 − 0.669i)3-s + (−0.345 − 0.938i)4-s + (0.974 − 0.226i)6-s + (0.967 + 0.254i)8-s + (0.104 + 0.994i)9-s + (−0.371 + 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (−0.875 − 0.483i)18-s + (0.217 + 0.976i)19-s + (−0.971 − 0.235i)23-s + (−0.548 − 0.836i)24-s + (0.161 + 0.986i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.572 + 0.820i)2-s + (−0.743 − 0.669i)3-s + (−0.345 − 0.938i)4-s + (0.974 − 0.226i)6-s + (0.967 + 0.254i)8-s + (0.104 + 0.994i)9-s + (−0.371 + 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (−0.875 − 0.483i)18-s + (0.217 + 0.976i)19-s + (−0.971 − 0.235i)23-s + (−0.548 − 0.836i)24-s + (0.161 + 0.986i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4028835609 + 0.2171496633i\)
\(L(\frac12)\) \(\approx\) \(0.4028835609 + 0.2171496633i\)
\(L(1)\) \(\approx\) \(0.5167345789 + 0.06791028770i\)
\(L(1)\) \(\approx\) \(0.5167345789 + 0.06791028770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.572 + 0.820i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.717 - 0.696i)T \)
17 \( 1 + (-0.508 + 0.861i)T \)
19 \( 1 + (0.217 + 0.976i)T \)
23 \( 1 + (-0.971 - 0.235i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (-0.532 + 0.846i)T \)
37 \( 1 + (-0.556 - 0.830i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (-0.875 + 0.483i)T \)
53 \( 1 + (-0.647 + 0.761i)T \)
59 \( 1 + (-0.797 - 0.603i)T \)
61 \( 1 + (-0.905 - 0.424i)T \)
67 \( 1 + (-0.189 + 0.981i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (-0.475 - 0.879i)T \)
79 \( 1 + (0.625 + 0.780i)T \)
83 \( 1 + (0.791 - 0.610i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (-0.0570 - 0.998i)T \)
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

−18.20385159197243989662113080126, −17.50814111148606606669362575300, −16.790899953476373600404010899968, −16.24517387456924498098302550742, −15.676683925641235967093062289130, −14.844748964403380887623436044255, −13.71841160026975595185400524711, −13.32399439493168745319488113861, −12.36690826344328881411144446667, −11.54064665082437435078145686967, −11.37677741383663226420715463361, −10.66689066671694466583084834354, −9.75325100673342487748809863397, −9.37874312389496618366435213537, −8.72389019931849720717959171925, −7.78296320398157114045357694541, −6.90868129577638171096355039568, −6.24081858000585584671148075102, −5.16786787833453425645663739472, −4.52682963273723345060776544447, −3.81474695192654405225497099538, −3.11533948229883399748573979034, −2.08140700338616310140311968502, −1.21729142519501476347301551962, −0.21941547870009727415238031748, 0.37646650359327113312895462877, 1.59696692910812073400165428415, 1.83032142880810804659579492594, 3.41444671031888528852112885608, 4.34462558419934556662595317807, 5.29479115815644915593445771914, 5.89152953022108580143826746808, 6.31566005787704910173345629890, 7.20447136545241572552857612028, 7.82947414389389529017807058701, 8.40045861694172805125995339719, 9.16354305955158583349834156410, 10.245880369510568147773192092889, 10.63338603194808459088989274533, 11.26607057391877509162757764910, 12.349004211790743833054567848739, 12.800620663413503133316922330212, 13.720469291379380321772855630818, 14.174327749207653637386569550263, 15.12883956967136792867271225238, 15.78875053596114195431387210444, 16.39474678432864014957926338014, 16.99994286586265862258046585794, 17.69146231785661718679868706053, 18.15734038095988084365583184848

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