Properties

Degree 1
Conductor 23
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 24-s + 25-s + 26-s + 27-s − 28-s + 29-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 24-s + 25-s + 26-s + 27-s − 28-s + 29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{23} (22, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.455362513$
$L(\frac12,\chi)$  $\approx$  $2.455362513$
$L(\chi,1)$  $\approx$  1.965202054
$L(1,\chi)$  $\approx$  1.965202054

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

−38.68724240329872765497576852621, −37.96791783033085224078040530246, −36.07573583369179760813970017175, −34.87017245956883141460396113366, −33.15130794249205310083365342979, −31.91833869373659697245162297098, −31.25469395122993651116980291877, −30.10171925835832877824003242347, −28.5028654987024245497552475716, −26.48639123783683881594192774017, −25.462561010482497667429215273674, −23.94644729526781844527668854668, −22.87407602346311971158763595981, −21.20360225639625471304374327961, −19.98776684788736769840436487917, −18.98593120269587444083248143723, −15.93810136734798316256870844575, −15.38556073359858098799258622111, −13.60413172473855533286025588645, −12.581696786969624525139642734401, −10.63387123021858563292939521079, −8.334849030124398104335730457447, −6.731189150719542313772896829694, −4.21518980422971907980891242449, −2.871339848930367916555243647734, 2.871339848930367916555243647734, 4.21518980422971907980891242449, 6.731189150719542313772896829694, 8.334849030124398104335730457447, 10.63387123021858563292939521079, 12.581696786969624525139642734401, 13.60413172473855533286025588645, 15.38556073359858098799258622111, 15.93810136734798316256870844575, 18.98593120269587444083248143723, 19.98776684788736769840436487917, 21.20360225639625471304374327961, 22.87407602346311971158763595981, 23.94644729526781844527668854668, 25.462561010482497667429215273674, 26.48639123783683881594192774017, 28.5028654987024245497552475716, 30.10171925835832877824003242347, 31.25469395122993651116980291877, 31.91833869373659697245162297098, 33.15130794249205310083365342979, 34.87017245956883141460396113366, 36.07573583369179760813970017175, 37.96791783033085224078040530246, 38.68724240329872765497576852621

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