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Properties

Label 1-23-23.9-r0-0-0
Degree $1$
Conductor $23$
Sign $0.986 - 0.165i$
Analytic cond. $0.106811$
Root an. cond. $0.106811$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(0.106811\)
Root analytic conductor: \(0.106811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (0:\ ),\ 0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8564154349 - 0.07119029211i\)
\(L(\frac12)\) \(\approx\) \(0.8564154349 - 0.07119029211i\)
\(L(1)\) \(\approx\) \(1.130623330 - 0.08441276772i\)
\(L(1)\) \(\approx\) \(1.130623330 - 0.08441276772i\)

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

−39.53057000057408974443827257399, −38.06405493719890636932453680584, −35.90801217877348028996675776859, −35.07041491708066181269011239110, −34.28831801930400569837137837857, −32.23632742777255278181244748430, −31.47207768496525831651942006288, −30.230316345376084729879785663673, −29.02871024721695166349870007671, −27.16701403565190250789603003861, −25.20173328033095296890873391070, −24.56382185594355984199241057903, −23.14452153449007311296374474522, −22.281227628668897155281978790810, −20.1602908688378450602602780520, −18.8444819164539367817621309389, −16.97064077543365217254061712072, −15.65507743925292049310893565467, −14.074306084298957911601624029266, −12.43425355313810107408547889512, −11.831741325069531222304676562549, −8.45117785217496318247771938581, −7.07465734175107573187136964239, −5.51944896175773567927746275915, −3.180067760838514227825587366893, 3.461093698942710071367139511604, 4.53726027325961264147771829576, 6.7760396547694166619140979147, 9.61449456153161740366302226097, 10.96302855097207220819127941529, 12.22204862691147626677748261347, 14.26452315674057170625267718594, 15.38415553822810622133363281295, 16.73628140793976823226951788309, 19.42429341204197976016823986381, 20.16997259280709505760435951272, 21.85239216769367598169346995665, 22.76816376270039974292210669320, 23.85940471327194475732402853122, 26.07174879497653635390264330688, 27.41593104953179932536090368156, 28.565267481620868950925252494262, 30.08605009406770738424560623997, 31.368129940671556856465118950469, 32.49414839088198040884830167161, 33.4264123054861695471883194583, 34.90206818040353173284656919011, 36.81912418317625834806522453641, 38.526479490849687529820230185270, 38.80237792185090640186982278813

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