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Properties

Label 1-421-421.199-r0-0-0
Degree $1$
Conductor $421$
Sign $0.722 - 0.691i$
Analytic cond. $1.95511$
Root an. cond. $1.95511$
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  + (0.936 − 0.351i)2-s + (0.858 − 0.512i)3-s + (0.753 − 0.657i)4-s + (0.753 + 0.657i)5-s + (0.623 − 0.781i)6-s + (0.858 + 0.512i)7-s + (0.473 − 0.880i)8-s + (0.473 − 0.880i)9-s + (0.936 + 0.351i)10-s + (−0.995 − 0.0896i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (0.983 + 0.178i)14-s + (0.983 + 0.178i)15-s + (0.134 − 0.990i)16-s + (−0.550 − 0.834i)17-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (0.858 − 0.512i)3-s + (0.753 − 0.657i)4-s + (0.753 + 0.657i)5-s + (0.623 − 0.781i)6-s + (0.858 + 0.512i)7-s + (0.473 − 0.880i)8-s + (0.473 − 0.880i)9-s + (0.936 + 0.351i)10-s + (−0.995 − 0.0896i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (0.983 + 0.178i)14-s + (0.983 + 0.178i)15-s + (0.134 − 0.990i)16-s + (−0.550 − 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.203652593 - 1.286798305i\)
\(L(\frac12)\) \(\approx\) \(3.203652593 - 1.286798305i\)
\(L(1)\) \(\approx\) \(2.415124854 - 0.7033020015i\)
\(L(1)\) \(\approx\) \(2.415124854 - 0.7033020015i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.936 - 0.351i)T \)
3 \( 1 + (0.858 - 0.512i)T \)
5 \( 1 + (0.753 + 0.657i)T \)
7 \( 1 + (0.858 + 0.512i)T \)
11 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.550 - 0.834i)T \)
19 \( 1 + (-0.963 + 0.266i)T \)
23 \( 1 + (-0.393 + 0.919i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.691 + 0.722i)T \)
37 \( 1 + (-0.995 - 0.0896i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.134 - 0.990i)T \)
47 \( 1 + (0.753 + 0.657i)T \)
53 \( 1 + (0.858 + 0.512i)T \)
59 \( 1 + (-0.0448 - 0.998i)T \)
61 \( 1 + (0.134 - 0.990i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.134 + 0.990i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (-0.691 - 0.722i)T \)
97 \( 1 + (-0.963 - 0.266i)T \)
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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

−24.25870176435916076069182557487, −23.77255152098586581220371853778, −22.39915776348859216661824945708, −21.546901605287165025023095146229, −21.009093335693729145187650958060, −20.344861547284423563410122722753, −19.62379336433117020244113236652, −17.96115419541072961892558006916, −17.11481866450626062020477311139, −16.34695097259372878793011037507, −15.20726619984703098164683853948, −14.73664572309268182033985827678, −13.70135865676038633190602384958, −13.15594573272407773972036975811, −12.28889478266362602683936561551, −10.66807117761299937349999950761, −10.25643746739725253493191384603, −8.604862621532435238046446690769, −8.13851292260589515193002057023, −7.02491573733697255730750281842, −5.59628655457911213568856810100, −4.762601155159540653501089090626, −4.14639614655243862025561186255, −2.610206750730345564741752399447, −1.942093651431370007062270885013, 1.82052364001952955063958355140, 2.286859724424004425064179550789, 3.22138994052706452866824013664, 4.64663224557109731036295527463, 5.58995436139210822833616713127, 6.7517433347877589222417050071, 7.49375746002113033860435503977, 8.7932878893106817128463738328, 9.88983432547803436291756036040, 10.800190131145450278539956149796, 11.88606246527229398424838174311, 12.71157653939828438522966241431, 13.83986072466821116425155671948, 14.08529361078940625712991187555, 15.06883681183626166918775608287, 15.69475115561339878120797931812, 17.37998558622651779502689661732, 18.34063958762618123168533821273, 18.93465543704613293078253527039, 19.92997362094097557057111071252, 20.93447833508592502349422908090, 21.41631130725308729679690490316, 22.117756186533521909299562379315, 23.47122496775407857057890843243, 23.98818904414053502239333127736

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