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Properties

Label 1-920-920.19-r0-0-0
Degree $1$
Conductor $920$
Sign $0.268 + 0.963i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (−0.841 + 0.540i)3-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (0.142 + 0.989i)27-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + (−0.415 + 0.909i)37-s + (0.959 − 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (0.142 + 0.989i)27-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + (−0.415 + 0.909i)37-s + (0.959 − 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.268 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8997533484 + 0.6829180363i\)
\(L(\frac12)\) \(\approx\) \(0.8997533484 + 0.6829180363i\)
\(L(1)\) \(\approx\) \(0.8787614783 + 0.2385384551i\)
\(L(1)\) \(\approx\) \(0.8787614783 + 0.2385384551i\)

Euler product

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

−21.897706818923126639548593981900, −21.15661914167706473099764498085, −19.960037404677611022721425734170, −19.315691222030725301701158538676, −18.36303469839451034875644677978, −17.80823845468994923415749360980, −17.07121671517131922321569240072, −16.33631214297297511254954790723, −15.45827350494003905374314572778, −14.24556910941343429952144626746, −13.902713858194360816345221329734, −12.62456269592093933836132491589, −12.01465978413810814490257307399, −11.21850531347325709070636184428, −10.749643706271081943288738582536, −9.354074648751612607441810725147, −8.65068154986860434571959959403, −7.36904984698493567897928359202, −7.029581808366843261935266649893, −5.756001191239597973196100048856, −5.11806964248809078130520069646, −4.2876877355809287513769541362, −2.75257284716876633763608123562, −1.77171846540194436712609509472, −0.64743798164629875188909300949, 1.13306794825294945138259348477, 2.17427306166622886095248705237, 3.83433664197392860888951060719, 4.384114611661692256503033531392, 5.29987692675783405926191707147, 6.12967590502814629269696062425, 7.184099318820795864866417895425, 7.9758849616342288155590338970, 9.15044784652275009822682525387, 10.06442536891042700029263196379, 10.593159688902130165841826090017, 11.663287327401730574452794695494, 12.11438832022080617903965157484, 13.04199406732324943017606686669, 14.4027647971910016716502650097, 14.84978903631973651889024487492, 15.60681030287102512531804583706, 16.87363132770910212124752598630, 17.13787607384851156849880606717, 17.83296312623268631037926520759, 18.75430043283927652950235299931, 19.86574308232508557982469392845, 20.58150455083892767146031840274, 21.30844403958945667052814243975, 22.11923314893115105645196191808

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