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Properties

Label 1-460-460.279-r1-0-0
Degree $1$
Conductor $460$
Sign $-0.948 + 0.316i$
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  + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.415 − 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.841 − 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.415 − 0.909i)33-s + (0.142 − 0.989i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)3-s + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.415 − 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.841 − 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.415 − 0.909i)33-s + (0.142 − 0.989i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ -0.948 + 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1088445591 - 0.6694069840i\)
\(L(\frac12)\) \(\approx\) \(-0.1088445591 - 0.6694069840i\)
\(L(1)\) \(\approx\) \(0.7172665033 - 0.3476989734i\)
\(L(1)\) \(\approx\) \(0.7172665033 - 0.3476989734i\)

Euler product

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

−24.1333178871936939878733853304, −23.3006172598330598688960507684, −22.19811198963826078715217424777, −21.68674512621406110327494891192, −21.1695862411279671275533478206, −19.87095182338882394459788259176, −19.09746679728053018591270031858, −17.95840746852814033566247916954, −17.24002912732776653324045349925, −16.50722331949921026654154541640, −15.428659593494983659407993477770, −14.90354805246218903613811464721, −13.90031318472129123915717013508, −12.516312707396967160362872227005, −11.70140514885008455834075563346, −11.1648983054757002747306632726, −9.99769069688003416665373953296, −9.05413565284842456312727250795, −8.43684356608875820589197080368, −6.63684468254895110501034802398, −6.17752275045000128258277587103, −4.80730790098523174708721698225, −4.28774092893500095570779405303, −2.85152047021959531525514739961, −1.49047866827953096378416682437, 0.205815754697906878213586138290, 1.25169852941668118988664398586, 2.40646763584742500109672884283, 4.0261663833479918652466135735, 4.93070732730621266502056017077, 6.11645711327869046210455240976, 7.00071817320539623105762971305, 7.72154127176049192574460666831, 8.808660036056870110294020316324, 10.221455011713245888176951987432, 10.881164297551952282811661875607, 11.87976381865888811701754882168, 12.62805207139228520637173887224, 13.612715392998594044461422793764, 14.332418166077282026173591991105, 15.45877687766718206760128451151, 16.676653718700929713264775758029, 17.32058033352203541765089352157, 17.83932088990958427453357529176, 18.94931352132409873363950703952, 19.8397530044225186047644135804, 20.44245311398804350522094861165, 21.76426205041142757267414793091, 22.60727969283674916666286646388, 23.188725652471121548879250114776

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