↑

Properties

Label 1-4560-4560.1067-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.999 + 0.0130i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0130i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.999 + 0.0130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.840650456 + 0.01196894737i\)
\(L(\frac12)\) \(\approx\) \(1.840650456 + 0.01196894737i\)
\(L(1)\) \(\approx\) \(1.129043194 - 0.04312867442i\)
\(L(1)\) \(\approx\) \(1.129043194 - 0.04312867442i\)

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

−18.333257338261744113503162215015, −17.44954724081895531433916379779, −16.992013004175865018147149391051, −16.06078852357052348086229286360, −15.65313496363246259144732196353, −14.930711110505171964513738239490, −14.21461032104113615177995884807, −13.45962551423069213463127326600, −12.78088628852004843430081806483, −12.15882320674870716789727209438, −11.60565575804274382694226844358, −10.671149678836680629112836333920, −9.92924977507283995984663477424, −9.405697270707642336012643445978, −8.71509258442419225122431902363, −7.85696286442666760524149757882, −7.20008841894328178956491235926, −6.1675153014865761002529157263, −5.94230405756634834758952887310, −4.99673280097509446851216800617, −3.89992099363121559132787456614, −3.47478807630552425689886866117, −2.57933844694768617536249088550, −1.63683663361793569986268722235, −0.7068047641150968999721014343, 0.814573666286813358020255222215, 1.42931509245418382433221549746, 2.706753841938487996492656210070, 3.38473077383949706820351028918, 4.05337807113739997755985850373, 4.77556515172206801788376729845, 6.02598105948132738760429639418, 6.28136314339155724761205114926, 7.068516823399151766519372719792, 7.8735274943476937234757118608, 8.77124738156116816059470216154, 9.2564097672871683060580159593, 10.08123681937810550942742365164, 10.68501739554492038238513369656, 11.44968207087479950772060788997, 12.21597118685821967216053160628, 12.741025152989381220249641941758, 13.696253611975089772813437428443, 14.09397797857840273789967730316, 14.694911046327456599769818147646, 15.801238164973724894644466326421, 16.297039196849961436364858535200, 16.71506602396660038383054392727, 17.43632484720934045665658735885, 18.464275860873135749020195428667

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