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Properties

Label 1-223-223.209-r1-0-0
Degree $1$
Conductor $223$
Sign $0.646 + 0.762i$
Analytic cond. $23.9646$
Root an. cond. $23.9646$
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  + (0.210 + 0.977i)2-s + (−0.0424 + 0.999i)3-s + (−0.911 + 0.411i)4-s + (0.594 − 0.803i)5-s + (−0.985 + 0.169i)6-s + (0.873 − 0.487i)7-s + (−0.594 − 0.803i)8-s + (−0.996 − 0.0848i)9-s + (0.911 + 0.411i)10-s + (−0.985 + 0.169i)11-s + (−0.372 − 0.927i)12-s + (−0.0424 − 0.999i)13-s + (0.660 + 0.750i)14-s + (0.778 + 0.628i)15-s + (0.660 − 0.750i)16-s + (0.985 − 0.169i)17-s + ⋯
L(s)  = 1  + (0.210 + 0.977i)2-s + (−0.0424 + 0.999i)3-s + (−0.911 + 0.411i)4-s + (0.594 − 0.803i)5-s + (−0.985 + 0.169i)6-s + (0.873 − 0.487i)7-s + (−0.594 − 0.803i)8-s + (−0.996 − 0.0848i)9-s + (0.911 + 0.411i)10-s + (−0.985 + 0.169i)11-s + (−0.372 − 0.927i)12-s + (−0.0424 − 0.999i)13-s + (0.660 + 0.750i)14-s + (0.778 + 0.628i)15-s + (0.660 − 0.750i)16-s + (0.985 − 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(223\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(23.9646\)
Root analytic conductor: \(23.9646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{223} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 223,\ (1:\ ),\ 0.646 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860766473 + 0.8621460879i\)
\(L(\frac12)\) \(\approx\) \(1.860766473 + 0.8621460879i\)
\(L(1)\) \(\approx\) \(1.106023204 + 0.6029461018i\)
\(L(1)\) \(\approx\) \(1.106023204 + 0.6029461018i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad223 \( 1 \)
good2 \( 1 + (0.210 + 0.977i)T \)
3 \( 1 + (-0.0424 + 0.999i)T \)
5 \( 1 + (0.594 - 0.803i)T \)
7 \( 1 + (0.873 - 0.487i)T \)
11 \( 1 + (-0.985 + 0.169i)T \)
13 \( 1 + (-0.0424 - 0.999i)T \)
17 \( 1 + (0.985 - 0.169i)T \)
19 \( 1 + (0.524 + 0.851i)T \)
23 \( 1 + (0.594 - 0.803i)T \)
29 \( 1 + (0.660 - 0.750i)T \)
31 \( 1 + (0.942 + 0.333i)T \)
37 \( 1 + (0.985 + 0.169i)T \)
41 \( 1 + (-0.721 + 0.691i)T \)
43 \( 1 + (-0.292 + 0.956i)T \)
47 \( 1 + (0.524 - 0.851i)T \)
53 \( 1 + (-0.967 - 0.251i)T \)
59 \( 1 + (-0.372 + 0.927i)T \)
61 \( 1 + (-0.0424 - 0.999i)T \)
67 \( 1 + (0.911 - 0.411i)T \)
71 \( 1 + (-0.372 + 0.927i)T \)
73 \( 1 + (-0.450 - 0.892i)T \)
79 \( 1 + (0.721 - 0.691i)T \)
83 \( 1 + (-0.292 - 0.956i)T \)
89 \( 1 + (-0.594 - 0.803i)T \)
97 \( 1 + (0.594 + 0.803i)T \)
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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

−26.13995276192407624704125078698, −25.17979938933817557567723575023, −23.84129916186750465065431155376, −23.498655736513085890303122702677, −22.17840079608041738946145997135, −21.43185582406525230105182356875, −20.63776934886960275145242897478, −19.2321524915666052550284462692, −18.67924032387282668747808888301, −17.97675681503149571982691420108, −17.19898623358269036033196112410, −15.210066654278276068069373755503, −14.10781100961625301704605257921, −13.694184136258540636097366004288, −12.50677954671883904993146654302, −11.52514321486207532611790024406, −10.8798196094948439044777923778, −9.57927371436599750124381558103, −8.42061309182474576683798014400, −7.25762686682516647619049181459, −5.85967138487356266870851289911, −5.01417244351749256589575024607, −3.07382781184546113795734771364, −2.26576767819353450991451726999, −1.20224656312291370774755792489, 0.74762204038703764676713929068, 3.05967652073137391002984928256, 4.577069345522239388998408575464, 5.102684743776741549795165030232, 6.00386297268754656987029411806, 7.88912867529590122551118515974, 8.338240135976760022844647483, 9.76760293813937591616203950781, 10.35945337176133346406022705251, 12.02200552724726582598535640037, 13.18776528666965390768496737431, 14.140105552144479585583244639567, 14.97745979330292946242378969913, 15.97793610721485034686928992756, 16.77169732463636855260309652966, 17.48452098846853197681715091144, 18.37605869285204061901234483549, 20.29687196766271804148604420285, 20.93899407320562237607866994587, 21.59205886922289799127520125083, 22.93074371639285154349018238851, 23.43062488987500974583615534208, 24.77042684571154337134162437295, 25.24910196442706659710058370785, 26.4464372195959609986041786095

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