↑

Properties

Label 1-504-504.13-r1-0-0
Degree $1$
Conductor $504$
Sign $-0.939 + 0.342i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
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)5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s − 53-s − 55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s − 53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1017011713 - 0.5767760043i\)
\(L(\frac12)\) \(\approx\) \(-0.1017011713 - 0.5767760043i\)
\(L(1)\) \(\approx\) \(0.7889903133 - 0.2871689891i\)
\(L(1)\) \(\approx\) \(0.7889903133 - 0.2871689891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
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 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - T \)
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

−23.93095960150912317663264031409, −22.8701795086254645806789327328, −22.323958314031203184622964250241, −21.56679559144348643368180178559, −20.32938474655932284126291475060, −19.64955416891741677762979232117, −18.89534716532248837242336033713, −17.89926491233615920341513543299, −17.29354461514270918520900704902, −15.978472298139352955059039144227, −15.40501722307531570829573237688, −14.37698587396853522795991425590, −13.83141898333962597295658208328, −12.4340579880664779871306558749, −11.718492987876626599964216660997, −10.933083898488724744140858084423, −9.82899602705689343500243469982, −9.090505357129877653507130596479, −7.694410856806767155269098990105, −7.068438959281139246050975149049, −6.20138233406975768843345924346, −4.75213037510449002097658097226, −3.91925237223173531350424983059, −2.75215877992200694686082350612, −1.65087672386635575014368522362, 0.16098002007585429376204943154, 1.14237729439001639665614552853, 2.69100026027831084455992385188, 3.84141787729221384952221042183, 4.81532844067520883572106624743, 5.75320601090620244545511371035, 6.91087911377718604995344272920, 8.07498248360166045941088600004, 8.67164733931808939073744900611, 9.6859216871549509689618438283, 10.78635777056036558174938447176, 11.78110847534750247232949025122, 12.4405927695980518302708861171, 13.447197939345560581314636267059, 14.2464010941235966296585459308, 15.51783506100311797712703196641, 15.9885955847386134996312562914, 17.007368072313585104065002840171, 17.7056595210972555365200816890, 18.84119918579319951243915752869, 19.781040224183440329184903704489, 20.22153611135468410672547745515, 21.230596784033117762775693868827, 22.20764445724185986184441115983, 22.873269045049689706378453691970

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