Properties

Degree 1
Conductor 23
Sign $0.987 + 0.159i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−0.841 + 0.540i)11-s + (0.841 − 0.540i)12-s + (−0.654 − 0.755i)13-s + (0.959 − 0.281i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + ⋯
L(s,χ)  = 1  + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−0.841 + 0.540i)11-s + (0.841 − 0.540i)12-s + (−0.654 − 0.755i)13-s + (0.959 − 0.281i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.987 + 0.159i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.987 + 0.159i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $0.987 + 0.159i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (5, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (1:\ ),\ 0.987 + 0.159i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.083085753 + 0.1672266969i$
$L(\frac12,\chi)$  $\approx$  $2.083085753 + 0.1672266969i$
$L(\chi,1)$  $\approx$  1.683741036 + 0.1252447533i
$L(1,\chi)$  $\approx$  1.683741036 + 0.1252447533i

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

−38.74397818811945791663464534861, −37.63438665308647743595581587069, −36.7946839313765054068397415341, −34.05165362945192283689215830256, −33.53372883901863086772922396536, −32.03221983563972448910809793722, −31.32375436378571313955777553146, −29.324395814452884084665761854139, −28.56662802267029037447180766571, −27.157031368684222388671121074695, −25.28896530532217233251280210055, −23.91896783424091893298494423174, −22.146580197257299225445900613472, −21.38467987556147591535047687090, −20.50746819469770569999969825314, −18.4276888619449538673689442386, −16.509337093457073672242413117537, −14.99691157482120151679073361859, −13.79395777511727040962942646010, −11.9600848508228963369632335014, −10.52705860820053999574560484821, −9.14729345802198967749456421469, −5.80015224206707440634641050469, −4.7463531721381654522177335807, −2.473712372400737954111502514698, 2.35411168788392419350667347488, 5.12010715496190400374515675553, 6.682122929086607135531999587529, 7.96669685476324552466378490646, 10.80333450597320314622754371375, 12.7144367997136803495128723353, 13.64357095106010376302118665678, 14.9267297498883991927605622614, 17.218062968277537192783677955175, 17.81718023429885344931949177912, 20.10092556287774272054688421257, 21.58826294349130332800064351903, 23.05211363346169218849256179827, 24.10433004235077031212481342476, 25.24581029958346973264782227137, 26.37364047768445097234497531551, 28.821038780962283511499050632948, 30.04958408653655997840449446567, 30.69616045890646108967806673004, 32.415047556969178546750986775205, 33.7368874145980544708760013327, 34.52186655028900444597055748881, 36.165657406880078154000483734925, 37.17654685153411594795120987538, 39.33482589629605334274113911229

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