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Properties

Label 1-460-460.443-r0-0-0
Degree $1$
Conductor $460$
Sign $-0.504 - 0.863i$
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  + (0.540 − 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.281 − 0.959i)13-s + (−0.989 + 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.989 − 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.281 − 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (0.281 − 0.959i)7-s + (−0.415 − 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.281 − 0.959i)13-s + (−0.989 + 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.989 − 0.142i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.281 − 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(2.13623\)
Root analytic conductor: \(2.13623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (0:\ ),\ -0.504 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7379382312 - 1.285433258i\)
\(L(\frac12)\) \(\approx\) \(0.7379382312 - 1.285433258i\)
\(L(1)\) \(\approx\) \(1.053099315 - 0.6260943215i\)
\(L(1)\) \(\approx\) \(1.053099315 - 0.6260943215i\)

Euler product

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

−24.50570000920381593935418607628, −23.26604033413368781004621816887, −22.02897138224213577014986880668, −21.88023556350840576879691128521, −20.85077151571878795204558739613, −19.94893447852829581229864358236, −19.3038112450608228917308693657, −18.2075888631090789123366821417, −17.25343295541020451462862053481, −16.33509177148997071000311941608, −15.30744755386743601967729068428, −14.91524140378334944536913034967, −13.95172444889200407166426683503, −12.92323341336201077212801647261, −11.66797848912399384648627370558, −11.14023626553925497122475834784, −9.665600447711456475221215799896, −9.24961905060703194231656246532, −8.386846668945318025771734800400, −7.16960985280552787220331103877, −6.017780890654975914598897256884, −4.70607542858342115495177454512, −4.23359389657905838910577171249, −2.69374467397540904883516183515, −1.97745869732471620761922458720, 0.77337303371886516611466676736, 1.89306607856979361158536606644, 3.231317204561186459098490712146, 4.07112309246605237614039270244, 5.590346346667356142128191939062, 6.63179745810471657219026411355, 7.48170660996923784083624222834, 8.31659489990502357978129554043, 9.1852882342312780689030403940, 10.475570098889042229985290909889, 11.28055948558303115548612916970, 12.48067786691483802665934095945, 13.130980094756980287069608260138, 14.18113884347360714121802050539, 14.543473784008645185712929812428, 15.848820981768553996294919739356, 16.96567737703395290074768347999, 17.654187484749954178458244780110, 18.504271939099805292288236596, 19.56150515013858470145409115057, 20.03646307831086368262326896366, 20.82602450161531108198233382685, 22.01527284234928157333802445949, 22.93574722743952459875583212444, 23.811495907700901320983897676196

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