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Properties

Label 1-460-460.439-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.378 - 0.925i$
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  + (−0.959 + 0.281i)3-s + (−0.142 − 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (0.654 + 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (−0.654 + 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.142 − 0.989i)33-s + (−0.841 + 0.540i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)3-s + (−0.142 − 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (0.654 + 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (−0.654 + 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.142 − 0.989i)33-s + (−0.841 + 0.540i)37-s + (0.142 + 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.378 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4664357027 - 0.6944682971i\)
\(L(\frac12)\) \(\approx\) \(0.4664357027 - 0.6944682971i\)
\(L(1)\) \(\approx\) \(0.7442551527 - 0.09816312355i\)
\(L(1)\) \(\approx\) \(0.7442551527 - 0.09816312355i\)

Euler product

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

−24.09914634313663551180980965019, −23.02295902141008151575120650971, −22.3781182155577335537761394340, −21.45627309146117905619342360648, −20.923570229178032263465485170659, −19.35598924694297056430234824421, −18.58377080379910471660035845072, −18.27745701679004632868769952540, −16.95823045767382269141129166944, −16.24525987715930452276130048884, −15.65583644831179813320314328938, −14.28943871341093814680262467294, −13.45369763870329910234104046940, −12.30476854119347482139184433594, −11.79471116657080287913746995416, −10.93444764573771955281403906006, −9.8433682379368602566117865358, −8.85943886437588134538307616640, −7.72873485733994552874585264169, −6.6820343897856590862174896987, −5.70016619466350875755133762449, −5.162970368603648657262779515634, −3.698693564360993417292131005324, −2.370663744929060811095741652641, −1.097070912786159055619886298646, 0.298248574428960022377677019917, 1.38318010170010468280459006392, 3.15058439534611977302416729710, 4.27307656914501745606780944426, 5.12347932872168921687723582037, 6.1492954058502691114379758078, 7.1809315006695327382995918229, 7.94964544440410250085174566290, 9.58407147553496023152073501059, 10.26041021616313158587551421020, 10.90843215532626567685993263563, 12.03307169027190724427157239809, 12.84849460053763272167693959537, 13.670645551831568807623785553452, 15.0563875047279897911996672701, 15.64706094781858800413208586778, 16.6969131117430154284654602261, 17.419630635925980789609086070569, 17.97305242151729758147537518790, 19.13206730656683695401108462774, 20.24623232589691612110844827943, 20.82327104720235954195640614268, 21.88892153880621310363881951133, 22.81104202060755051837103873804, 23.22867213978412126051855446009

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