↑

Properties

Label 1-4560-4560.3509-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.976 - 0.217i$
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  + (−0.5 − 0.866i)7-s + (0.866 + 0.5i)11-s + (−0.642 − 0.766i)13-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)23-s + (0.984 − 0.173i)29-s + (0.5 + 0.866i)31-s + i·37-s + (−0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.342 + 0.939i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.866 + 0.5i)11-s + (−0.642 − 0.766i)13-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)23-s + (0.984 − 0.173i)29-s + (0.5 + 0.866i)31-s + i·37-s + (−0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.342 + 0.939i)53-s + (0.984 + 0.173i)59-s + (−0.342 + 0.939i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.976 - 0.217i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (3509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.976 - 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.716143462 - 0.1885150806i\)
\(L(\frac12)\) \(\approx\) \(1.716143462 - 0.1885150806i\)
\(L(1)\) \(\approx\) \(1.086669020 - 0.09611154632i\)
\(L(1)\) \(\approx\) \(1.086669020 - 0.09611154632i\)

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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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

−18.53590077662742133640783788029, −17.28456417723207721438513116248, −17.07187833035198453057556218201, −16.23841456316253346140615608234, −15.594001603149518586280135029075, −14.785853216726442286849310336801, −14.401987040783409909972711531415, −13.51081427376000294561237600461, −12.75566131584728040799411091232, −12.14456959359162504478853242385, −11.59682782625853527893049074526, −10.83323558020924940764199262025, −9.90973100701415864918267543445, −9.36599482041822537221431300535, −8.64376304764937235427310020885, −8.14974014732907541093488755994, −6.86572214364584149374823254050, −6.59926407707413241630404631776, −5.731905498203781764136755890245, −5.021181313500170196335072393642, −4.07229610707449417848506491125, −3.4165895160889555447130508226, −2.48974853603852578652350609493, −1.81879310310989710794515263112, −0.67938072287198125614868375183, 0.76072546350247169410786566558, 1.39823735068163232314688626754, 2.77823718425158399666784524229, 3.14260541524130528817254560301, 4.22831732638681469454801741457, 4.77078410260253481726515582408, 5.60431744371548825985919476367, 6.70716725158087221580143907554, 6.97312160006779612444662169552, 7.76367548992481891414965888046, 8.59611727354579167081260740334, 9.569208263868882984203512083352, 9.86301197085235782510332956735, 10.665732577105814335983093963363, 11.420152994351655954385878789635, 12.24268723502339269257143820233, 12.69017124916649165714673922752, 13.64506596433709412433164683051, 14.04497485796958451515440613314, 14.84507285692521355323653576488, 15.560468219299065314129438537098, 16.18644162867021487934235710676, 17.1118825292777396205052848618, 17.294575990688425689687098086830, 18.099832655006402955605079757206

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