↑

Properties

Label 1-24-24.11-r0-0-0
Degree $1$
Conductor $24$
Sign $1$
Analytic cond. $0.111455$
Root an. cond. $0.111455$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯
L(s)  = 1  + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.111455\)
Root analytic conductor: \(0.111455\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24} (11, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 24,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7094580614\)
\(L(\frac12)\) \(\approx\) \(0.7094580614\)
\(L(1)\) \(\approx\) \(0.9358813101\)
\(L(1)\) \(\approx\) \(0.9358813101\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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

−38.757045413520358200019297607464, −37.31777594996008237908433584673, −36.33074388995123336856075573360, −34.88719921127767034411870622928, −33.49155789395814591322037091025, −32.42353686755135030793518084391, −31.08725180031803397862031783470, −29.21969083370577778290256668488, −28.86856257427073136901468159166, −26.7561074123782310792383080900, −25.67129733212202220561359782065, −24.40914964223159369359819602153, −22.66785732205079621070293214728, −21.55547824218793740139608072893, −20.034611576750188294726677609, −18.46329885012140044080926945086, −17.09898808691940348689297548911, −15.60365697037025378285194033903, −13.77715270577349586517982430976, −12.631153718166510585493030756588, −10.445721518412924823018776216323, −9.22463743993556115081806332595, −6.97192424379172752833414992011, −5.29243117677719815159648018861, −2.688658132467597586674142549469, 2.688658132467597586674142549469, 5.29243117677719815159648018861, 6.97192424379172752833414992011, 9.22463743993556115081806332595, 10.445721518412924823018776216323, 12.631153718166510585493030756588, 13.77715270577349586517982430976, 15.60365697037025378285194033903, 17.09898808691940348689297548911, 18.46329885012140044080926945086, 20.034611576750188294726677609, 21.55547824218793740139608072893, 22.66785732205079621070293214728, 24.40914964223159369359819602153, 25.67129733212202220561359782065, 26.7561074123782310792383080900, 28.86856257427073136901468159166, 29.21969083370577778290256668488, 31.08725180031803397862031783470, 32.42353686755135030793518084391, 33.49155789395814591322037091025, 34.88719921127767034411870622928, 36.33074388995123336856075573360, 37.31777594996008237908433584673, 38.757045413520358200019297607464

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