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Properties

Label 1-23-23.14-r1-0-0
Degree $1$
Conductor $23$
Sign $0.987 - 0.159i$
Analytic cond. $2.47169$
Root an. cond. $2.47169$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(2.47169\)
Root analytic conductor: \(2.47169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (1:\ ),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.083085753 - 0.1672266969i\)
\(L(\frac12)\) \(\approx\) \(2.083085753 - 0.1672266969i\)
\(L(1)\) \(\approx\) \(1.683741036 - 0.1252447533i\)
\(L(1)\) \(\approx\) \(1.683741036 - 0.1252447533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.959 + 0.281i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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