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Properties

Label 1-460-460.403-r0-0-0
Degree $1$
Conductor $460$
Sign $0.986 - 0.164i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9549285178 - 0.07927791460i\)
\(L(\frac12)\) \(\approx\) \(0.9549285178 - 0.07927791460i\)
\(L(1)\) \(\approx\) \(0.8653565817 + 0.09938254513i\)
\(L(1)\) \(\approx\) \(0.8653565817 + 0.09938254513i\)

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

−23.68947469925397358344247942059, −23.12406874463977060419859790057, −22.63956249670255182273992557787, −21.386475630997656361317559496968, −20.393907927374154535252746656970, −19.564362775760390887829321190540, −18.60908588846515770718750445003, −18.2388041398303206531683558795, −17.05450729148586954561225323390, −16.3057307933955778632161386424, −15.36993026541428153070334992967, −14.15452124039111318927769990306, −13.34887268612926071108648713292, −12.519008040123114830273534866308, −11.95458452103563847125951193745, −10.70601330035713993519285951264, −9.821656030052082328457189403482, −8.63873987867467474293477055774, −7.70003330963304540105223935407, −6.69583161088116541233737436671, −6.054781663142416273417330840039, −4.90615527040762960323689193386, −3.423044845275082621985784912183, −2.380955613729430221647778291, −1.10998079724180501973457664545, 0.66873565506781924925733028437, 2.79358672003807576628784710721, 3.57865075161487472933592729499, 4.55641526091112250621380802573, 5.95675034487110441564882105665, 6.25839130335135912160062123227, 7.96323770939533023519533770836, 8.87457211577343269072901152818, 9.79341366285962792219227779746, 10.65102596567887517481146470058, 11.29970505943974558383211861444, 12.55614029091603494912229264213, 13.42915358369468729341760026337, 14.40014789783799455961537766148, 15.59774698669474127037931660604, 15.985841141997682612281419710757, 16.84855793572175310284222997559, 17.69924767121418483288725233370, 18.960003917602522632508590621042, 19.55943858093453923668810915819, 20.7483504792067698954892800857, 21.3572402253799925154078137926, 22.11041074544487984271308453163, 23.14735824581799655772441700957, 23.48521663921725458268657440556

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