Properties

Label 1-421-421.78-r0-0-0
Degree $1$
Conductor $421$
Sign $0.0527 + 0.998i$
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  + (0.753 + 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 + 0.990i)4-s + (0.134 − 0.990i)5-s + (−0.222 + 0.974i)6-s + (0.473 − 0.880i)7-s + (−0.550 + 0.834i)8-s + (−0.550 + 0.834i)9-s + (0.753 − 0.657i)10-s + (0.983 − 0.178i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.936 − 0.351i)14-s + (0.936 − 0.351i)15-s + (−0.963 + 0.266i)16-s + (−0.393 − 0.919i)17-s + ⋯
L(s)  = 1  + (0.753 + 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 + 0.990i)4-s + (0.134 − 0.990i)5-s + (−0.222 + 0.974i)6-s + (0.473 − 0.880i)7-s + (−0.550 + 0.834i)8-s + (−0.550 + 0.834i)9-s + (0.753 − 0.657i)10-s + (0.983 − 0.178i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.936 − 0.351i)14-s + (0.936 − 0.351i)15-s + (−0.963 + 0.266i)16-s + (−0.393 − 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0527 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0527 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.0527 + 0.998i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.0527 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.829613244 + 1.735549956i\)
\(L(\frac12)\) \(\approx\) \(1.829613244 + 1.735549956i\)
\(L(1)\) \(\approx\) \(1.637785373 + 0.9813806805i\)
\(L(1)\) \(\approx\) \(1.637785373 + 0.9813806805i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.753 + 0.657i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (0.134 - 0.990i)T \)
7 \( 1 + (0.473 - 0.880i)T \)
11 \( 1 + (0.983 - 0.178i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.393 - 0.919i)T \)
19 \( 1 + (0.858 + 0.512i)T \)
23 \( 1 + (-0.691 + 0.722i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.0448 + 0.998i)T \)
37 \( 1 + (0.983 - 0.178i)T \)
41 \( 1 + (0.858 + 0.512i)T \)
43 \( 1 + (-0.963 + 0.266i)T \)
47 \( 1 + (0.134 - 0.990i)T \)
53 \( 1 + (0.473 - 0.880i)T \)
59 \( 1 + (-0.995 - 0.0896i)T \)
61 \( 1 + (-0.963 + 0.266i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.963 - 0.266i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (-0.0448 - 0.998i)T \)
97 \( 1 + (0.858 - 0.512i)T \)
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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

−24.02468469987803070659094243381, −22.98067734512712071830678386864, −22.24209546163113198014198463916, −21.6237663531126122402904237261, −20.39683516351104719836508294144, −19.76823750817783803701906154523, −18.88823378266813374187544114292, −18.182192366404461835346248845778, −17.54856790923958360218206924131, −15.55179962996355872177809530933, −14.896808119792535321620536069334, −14.273087557094733974830411591467, −13.433548685793294120190871240378, −12.43618946767773385573083515476, −11.72861111202194244193475985321, −10.902008221358939769931948839598, −9.73299327716299441613682186265, −8.68691913492417009139104381223, −7.51362747783250576718088188409, −6.279247340594820663649579725620, −5.89489025397688351991420387536, −4.24410546427849548698680649089, −3.05225983852889033310897057694, −2.38210096077475340810193910375, −1.31027049867655754900480530612, 1.61914423784042823010538865962, 3.29548299542690821603833920166, 4.26860043588013024097562998498, 4.71511382671953441182251913536, 5.85151876795512401347306453812, 7.13059989746272443802041580242, 8.17004193734392883509941642630, 8.97177747633002692395620205213, 9.82238864104166058713906134321, 11.36543697716954753469327279264, 11.925582235431205908548962789879, 13.449215625029460856924842735217, 13.93893633585676536166302742543, 14.54723101865726238143554848088, 15.907033646699530178637961318935, 16.33402235269732767644997616676, 17.00821572539599689128408537856, 17.9729468078423304905064620644, 19.96730893369142307101183392613, 20.07176917652141186576875458524, 21.3093554347871565565188366611, 21.55982749065152303868392468217, 22.79893545025705372496882117600, 23.54042479969757331263689541942, 24.59292469545385657898203289569

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