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Properties

Label 1-920-920.163-r0-0-0
Degree $1$
Conductor $920$
Sign $0.199 - 0.979i$
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  + (−0.281 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.199 - 0.979i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.199 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212359034 - 0.9903414512i\)
\(L(\frac12)\) \(\approx\) \(1.212359034 - 0.9903414512i\)
\(L(1)\) \(\approx\) \(1.062403054 - 0.4265603348i\)
\(L(1)\) \(\approx\) \(1.062403054 - 0.4265603348i\)

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

−22.017558116711365744743989599852, −21.22711721074019245897171558680, −20.41002701783380794815462457347, −20.21307194784394070520469997351, −18.7449880672098360273457494366, −17.8110375625916298678296891346, −17.50272448487275179201534399435, −16.31146149263951654797848985112, −15.79961210589764150771647197546, −14.81710685495609691972349384924, −14.3849660937884350050542092043, −13.31253241990529364151815019102, −12.11746407844104136574737433226, −11.43711085070882997988935525096, −10.80961078438090615652609633678, −9.83746595302906716824867490520, −9.08002985183481592584007374276, −8.266609036236352191729064439279, −7.221351944389489770341386369, −6.08029918120039030296782503545, −5.23319309140436850905048723293, −4.42684411659654268649271134942, −3.68790683656964578947275978253, −2.406667770944598223429285726268, −1.188161405987084497155414912195, 0.86928163425187815239942504859, 1.665300837262238845754051814892, 2.79777542203863071103217918503, 4.02046468149183105775017502925, 5.08595665261627514396066626019, 6.08210693631944312347929330414, 6.66991119855311810179344911348, 7.85752951537291806230422035434, 8.35373030120086467770796220109, 9.23423772172688836044508159652, 10.768598005233637544052735279555, 11.28056255421363143754811122880, 11.83213906106692960096385932077, 13.04596446640443842233216351352, 13.64673288079038654552860651331, 14.25540165677711114373416805931, 15.32217722977648466491714564203, 16.26899375798768042229271851039, 17.26204635562817102399553491379, 17.67343704849608327073833101325, 18.57442161261305693109641302789, 19.18538257318033041030349912011, 20.08045035624390546685639586393, 20.85683479168365057966160764715, 21.809544623715303706031641164514

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