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Properties

Label 1-460-460.163-r0-0-0
Degree $1$
Conductor $460$
Sign $-0.588 + 0.808i$
Analytic cond. $2.13623$
Root an. cond. $2.13623$
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  + (0.281 + 0.959i)3-s + (0.989 − 0.142i)7-s + (−0.841 + 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.989 − 0.142i)13-s + (−0.755 + 0.654i)17-s + (−0.654 + 0.755i)19-s + (0.415 + 0.909i)21-s + (−0.755 − 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)3-s + (0.989 − 0.142i)7-s + (−0.841 + 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.989 − 0.142i)13-s + (−0.755 + 0.654i)17-s + (−0.654 + 0.755i)19-s + (0.415 + 0.909i)21-s + (−0.755 − 0.654i)27-s + (0.654 + 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.989 − 0.142i)33-s + (−0.540 − 0.841i)37-s + (−0.142 − 0.989i)39-s + (0.841 + 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5540446505 + 1.088703967i\)
\(L(\frac12)\) \(\approx\) \(0.5540446505 + 1.088703967i\)
\(L(1)\) \(\approx\) \(0.9455077774 + 0.5249916757i\)
\(L(1)\) \(\approx\) \(0.9455077774 + 0.5249916757i\)

Euler product

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

−23.95892252219345078170431904537, −22.93597176972188796451168850613, −21.84417857083324697498577332984, −21.0373323843094331521162264145, −20.11091259824551154093162151575, −19.238810821419893237256080355236, −18.57137430754858010904664825239, −17.5784989543917910870285855909, −17.12000024039952703171260450239, −15.67332068215113420674836040782, −14.82685764015691868466472954951, −13.856196761232476749620928191867, −13.35720447998730326926506476984, −12.088804040727288068222037211772, −11.53571570569024930958689797234, −10.50185542103751080910623542890, −9.0445089919575153729210619341, −8.36355427534849763702685659550, −7.48410642206717714746804711413, −6.5787420413199431313497746325, −5.44214946349834238675147640695, −4.42922141091523369344677919939, −2.789662065190634927349323242801, −2.12520464050594279384635612808, −0.630806308232738512287193036944, 1.829003655205742992411684054133, 2.829344409414982051196741252917, 4.37411656782921947074282857185, 4.67575299547206049050274130743, 5.89567277673739333518727795839, 7.370884022324900182503260721815, 8.20206677987445598365615394766, 9.11299598706772745805303453811, 10.25241538311667404868194258347, 10.68894701537630361019870881120, 11.86541231066114087763536709918, 12.816272360226061834749430559545, 14.12207006339568731830619048682, 14.76808230317634377493626780398, 15.37263135634865419279126217533, 16.43107258030340675633682044, 17.4086640914748970256231755640, 17.9041904665443496937829765298, 19.421863426092992150574653819974, 19.99945361429020498992626794404, 21.0409494161460145104634585263, 21.38514440690264694492011874023, 22.486342002889325492515366433751, 23.222966575917594638818660628622, 24.30590868681688866966241032770

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