↑

Properties

Label 1-2952-2952.1229-r1-0-0
Degree $1$
Conductor $2952$
Sign $-0.984 - 0.173i$
Analytic cond. $317.236$
Root an. cond. $317.236$
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)7-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 − 35-s − 37-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-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 − 35-s − 37-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(317.236\)
Root analytic conductor: \(317.236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2952} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2952,\ (1:\ ),\ -0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1881405212 - 2.150455997i\)
\(L(\frac12)\) \(\approx\) \(0.1881405212 - 2.150455997i\)
\(L(1)\) \(\approx\) \(0.9758869678 - 0.5397918690i\)
\(L(1)\) \(\approx\) \(0.9758869678 - 0.5397918690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
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

−19.238293853978776008243905404051, −18.4497190035772005663049673049, −18.0847401799855683404625616526, −17.253216610951255651459341142389, −16.33790337446621710522875141745, −15.74884878783040055817494203993, −14.77497377190204468663540749755, −14.555918056058333403031604421909, −13.966645631382729271047902627789, −12.583159365447739268918531768, −12.09569969645194150271653165460, −11.62244415826381709981116023788, −10.76350019467702521637716957965, −9.99014399781766098896248209101, −9.23721246565358186740685283937, −8.50116006770796346688154417451, −7.55561048170999249508637373440, −7.028334187737357016864888950655, −6.32051840969958486277346398308, −5.21677771946187813279380789090, −4.675442305539018242464154053321, −3.623361957076245362371604795024, −2.882945087732699722777788827135, −2.04171040234479059872867587663, −1.1599533898417374382921108667, 0.42815109538979906886397985851, 0.88405139375889861040255220822, 1.77476194375735303562928686148, 3.31345300689635847132205571452, 3.60318499297419773263170966575, 4.72563873735905451710044032960, 5.27099222233792928427752727745, 6.06149621776968536747737050961, 7.307030918880546823192977292777, 7.77670136007722087238167970011, 8.32967806038563360900477795657, 9.360440051164559633454587773760, 9.891400025900309487680634235031, 10.941541625047168021589066831935, 11.46791078842353627787074078314, 12.221506769540059561116763399911, 12.89712408840011844735949794500, 13.81912194705516289966049214729, 14.15169899120038066234718568064, 15.227283212937960622500597328876, 15.80611738069746023073166068992, 16.64762875396165322793842287135, 17.123874782901966610399307400605, 17.64907705243402305536863563846, 18.750801963258618482213842144301

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