↑

Properties

Label 1-4560-4560.2837-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.880 - 0.474i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.766 + 0.642i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (0.939 + 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)11-s + (−0.766 + 0.642i)13-s + (−0.984 − 0.173i)17-s + (−0.342 + 0.939i)23-s + (0.984 − 0.173i)29-s + (−0.5 − 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.984 − 0.173i)47-s + (0.5 − 0.866i)49-s + (0.939 + 0.342i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.880 - 0.474i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.880 - 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04139197770 + 0.1638845146i\)
\(L(\frac12)\) \(\approx\) \(-0.04139197770 + 0.1638845146i\)
\(L(1)\) \(\approx\) \(0.7860319279 + 0.1476669026i\)
\(L(1)\) \(\approx\) \(0.7860319279 + 0.1476669026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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

−17.63503986174393595849109074668, −17.249929273685831033022569921830, −16.45395968889204587355204666466, −15.933537632394951113318317960626, −15.2026876226813623054387794461, −14.36242919331660358679054969677, −13.86328093163406714402694585080, −13.09896706650060697978190634994, −12.39506144981513452796364100180, −11.9295335150539497654018738092, −10.796756548394274489069360648464, −10.454396820217594119107115217663, −9.6584785460298295911584755912, −8.872933844302763619531297361711, −8.36964691911631956055816143788, −7.25033886478303070767464412867, −6.77470229780563096032639682000, −6.13878744165779027696682052432, −5.24050970447854093983350707852, −4.37179262415235866543606126813, −3.67996662772440085555418658849, −2.94154024384451117875083405055, −2.10757334480698157736131019668, −0.96744001106297712318418029328, −0.0507505135901881036811764494, 1.36145864080448912588281804567, 2.250727981591216472108900553893, 2.87560705753181623099671814515, 3.97113473616379468767337566482, 4.40831625891852811245749706488, 5.46231960263148157208777510817, 6.141577997610500427647595131273, 6.96888391086931676416115802304, 7.28001503819136320768469643642, 8.556831547557367834809834662104, 9.053418087620484189507775282383, 9.750220543361788989288375445482, 10.201189419270831324446019719489, 11.42758695001557923817080277083, 11.80828154397474988815243520581, 12.47014955590820831471750214873, 13.24948139691060728288699073973, 13.83649828819589090533411837916, 14.7172525660631516863166030185, 15.21607147553357699931426996488, 15.987192996154052923522171283420, 16.52942154190799860220879027091, 17.37919196545147980772818760547, 17.79410209924698006791985424322, 18.739641327461954040725675644594

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