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Properties

Label 1-460-460.447-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.845 + 0.534i$
Analytic cond. $49.4338$
Root an. cond. $49.4338$
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  + (−0.909 + 0.415i)3-s + (−0.540 + 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.540 − 0.841i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (0.142 − 0.989i)21-s + (−0.281 + 0.959i)27-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.540 + 0.841i)33-s + (0.755 + 0.654i)37-s + (0.841 + 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (−0.540 + 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.540 − 0.841i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (0.142 − 0.989i)21-s + (−0.281 + 0.959i)27-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.540 + 0.841i)33-s + (0.755 + 0.654i)37-s + (0.841 + 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.845 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1550994608 + 0.5355181270i\)
\(L(\frac12)\) \(\approx\) \(0.1550994608 + 0.5355181270i\)
\(L(1)\) \(\approx\) \(0.6760477637 + 0.1477704977i\)
\(L(1)\) \(\approx\) \(0.6760477637 + 0.1477704977i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (-0.540 + 0.841i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−23.30616535898399534416253690341, −22.64463729872529609224995801664, −21.93778424043099223900443224356, −20.75128495546597872955020299558, −19.88669906267951158620545503085, −19.02634036582825492951640886624, −18.0511619406766457752555223922, −17.40493715417093782124839374997, −16.42497822404875035581983426351, −15.96362669521735129690429637710, −14.53689611262881950319948424736, −13.61865568195840739040500375696, −12.77440215719210956961534354240, −11.92376610556755894358724996125, −11.13517195136538718407540853118, −9.990137402426295774734791889432, −9.4634002766960694270292426463, −7.59970069610019008917883943912, −7.177927198329909129002961687695, −6.24998447791650095358910770032, −4.988393173057791698659395290976, −4.28406496622277885588290141859, −2.71862646495628510204002386702, −1.364627009852720628096263854879, −0.20055255589019277648418379954, 1.06139213786916361823798998162, 2.81730136189657022583207143144, 3.75040119963651939432218859500, 5.217305227324681926599779220223, 5.734681594217715244767715595453, 6.65295093484825347552552788826, 7.98949062682305906084456780477, 9.04092731063096147541744606482, 10.06369640902703393989963046644, 10.72453415422089511824748437368, 11.90820031343521720672768766675, 12.40273108929526726773568670992, 13.45085954375563415095324283319, 14.746268668281873019603127490209, 15.60377150155287124142418564584, 16.233549000328078164999658080374, 17.11981783587886046465482635092, 18.02197468793227750094279446061, 18.80091019054474941991487922870, 19.70852399547846019453917176004, 20.88307155251530115446975194757, 21.83503144577058210243312954901, 22.13661602787296161703952972530, 23.119323353102165380120340214218, 23.983127323742744133961352446286

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