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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(24\)    =    \(2^{3} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{24} (5, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 24,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.571915159$
$L(\frac12,\chi)$  $\approx$  $1.571915159$
$L(\chi,1)$  $\approx$  1.282549830
$L(1,\chi)$  $\approx$  1.282549830

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

−38.14247751933542285576614257138, −37.169059080332133559649624087507, −36.06433119967803861641008437474, −34.34563209016533977564689216556, −33.37149707342212421697214612912, −32.11095898713639789271857905798, −30.4826325439914416234779210549, −29.47042426859464920571783262335, −27.99747354563209215912756582706, −26.69287992206634801816628273213, −25.114013937714922427003144983077, −24.18502420176868196704264889019, −22.24400758994810938440801208316, −21.23481315500759825470276937857, −19.7240153244415519968464947702, −17.87462416651816148400063708409, −17.030405571434057892804535988390, −14.88654143969881284582619693881, −13.75222966580168075032909065230, −11.935859369599332562538818607605, −10.231110375567980986681669522485, −8.631859564899361390761044162516, −6.55891715947090437049580839541, −4.72270950581234428327684375536, −1.97719051443795299435619611950, 1.97719051443795299435619611950, 4.72270950581234428327684375536, 6.55891715947090437049580839541, 8.631859564899361390761044162516, 10.231110375567980986681669522485, 11.935859369599332562538818607605, 13.75222966580168075032909065230, 14.88654143969881284582619693881, 17.030405571434057892804535988390, 17.87462416651816148400063708409, 19.7240153244415519968464947702, 21.23481315500759825470276937857, 22.24400758994810938440801208316, 24.18502420176868196704264889019, 25.114013937714922427003144983077, 26.69287992206634801816628273213, 27.99747354563209215912756582706, 29.47042426859464920571783262335, 30.4826325439914416234779210549, 32.11095898713639789271857905798, 33.37149707342212421697214612912, 34.34563209016533977564689216556, 36.06433119967803861641008437474, 37.169059080332133559649624087507, 38.14247751933542285576614257138

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