Properties

Label 1-460-460.3-r0-0-0
Degree $1$
Conductor $460$
Sign $0.768 + 0.640i$
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  + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.768 + 0.640i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.768 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131323228 + 0.4096645956i\)
\(L(\frac12)\) \(\approx\) \(1.131323228 + 0.4096645956i\)
\(L(1)\) \(\approx\) \(0.9677973522 + 0.2156906400i\)
\(L(1)\) \(\approx\) \(0.9677973522 + 0.2156906400i\)

Euler product

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

−23.80954673441708449986854565765, −23.113466361018630302059796142112, −22.1320946113666729068776488967, −21.534555879189355871063185462405, −20.24446611196264950785458250190, −19.55218600096365451228911098405, −18.54531176334088987342775930356, −17.75564939453876562935283503919, −16.97580595421901522658382573951, −16.47776147353159346759954047858, −14.94680044840307409607562016143, −14.26920653283905111076281282810, −13.30147038714153158975678793867, −12.22432003138162593690395606876, −11.66560798228457038450021839534, −10.71850237248778886599106896047, −9.8237998497547826783168900275, −8.35627378709693603503942800548, −7.609071931289041503561015779762, −6.70921428610022774482956540224, −5.614668627304480863120136876500, −4.81584136535014536530660263545, −3.55024498324236310540730870207, −1.91404266692714393037663743582, −1.01167737338526899259834013484, 1.09026599900881623801965333207, 2.56387712402841649950710618548, 4.03837170975589990148610026684, 4.848744529912726992616328422879, 5.64934848640094328879920654643, 6.81967030228300658926628653659, 7.799059822748071918829255636062, 9.29367472884245864821713665067, 9.617968008321789537609761461541, 10.96552252532648910082227646065, 11.80173161015665886114391053718, 12.11056907188793709704359382326, 13.71106302527207414798447976965, 14.71371864417809883541198318506, 15.24540300959823471960239797640, 16.419039775638133426232730276161, 17.092217675090483461348935487309, 17.85231924597724718896032126529, 18.69270683967238046240760310228, 19.91929083530694044905703421096, 20.774708572420228673418624919726, 21.55343473543387671079470887371, 22.32722005651540803736885426199, 22.91863702692075357769100929666, 24.2052283353943146231828431920

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