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Properties

Label 1-460-460.43-r1-0-0
Degree $1$
Conductor $460$
Sign $0.950 + 0.311i$
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  + (−0.755 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(49.4338\)
Root analytic conductor: \(49.4338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 460,\ (1:\ ),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5287867516 + 0.08450249616i\)
\(L(\frac12)\) \(\approx\) \(0.5287867516 + 0.08450249616i\)
\(L(1)\) \(\approx\) \(0.5650256763 + 0.08410578287i\)
\(L(1)\) \(\approx\) \(0.5650256763 + 0.08410578287i\)

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

−23.60531777438910702233932903855, −22.78661037886400784537340762439, −22.21345379178617159051256913474, −21.30710063218843905435736805911, −20.029436525926115259685175865691, −19.298316106716458737215417456997, −18.44898800083410683664885058933, −17.73961068819448689865749296656, −16.85106606184187539773840567410, −15.92423482921375616460217707498, −15.22397694010893934187686039616, −13.837775667497432819330244692235, −12.85611367510472139262608597108, −12.52317195395554522895443829537, −11.39958124542845940777030199446, −10.46142646165497404525064816401, −9.58028316417547898681729049174, −8.33699975368634320366516850531, −7.22653502342542757829689992449, −6.58905964689536616372676891530, −5.43564459761670297898155791515, −4.73619596198388174652319825592, −2.957837406932802969321784938909, −2.12498093804797364890140616293, −0.40666184054845363336632771496, 0.37778841836204229250805652233, 2.25670846741678300159555736114, 3.62667333325846968894144552465, 4.4340873959368777468477300125, 5.60538624589767070980832229851, 6.396702236570297961095085714537, 7.41910860993058077323790515767, 8.73973083971228158476472241714, 9.829792537062959539832147656375, 10.39914830004377058049157666149, 11.2598017852481615173496567973, 12.438784950321474327994375364267, 13.025583268663830986478133760630, 14.271132039209651973659295283528, 15.38307871172184794585413759715, 15.930994571138959271500805135111, 17.03109136087234643370244136299, 17.30011530073545382773593421328, 18.74129898208668678794828235827, 19.34147806055090620053645199827, 20.57465929847902210341931990447, 21.239124775889033260631837471369, 22.21092227769912466340236429496, 22.72882084436578320615653308781, 23.730725474266399870610427469299

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