Properties

Degree 1
Conductor $ 2^{3} \cdot 3 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{24} (11, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 24,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7094580614$
$L(\frac12,\chi)$  $\approx$  $0.7094580614$
$L(\chi,1)$  $\approx$  0.9358813101
$L(1,\chi)$  $\approx$  0.9358813101

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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