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Properties

Label 1-184-184.29-r0-0-0
Degree $1$
Conductor $184$
Sign $0.438 - 0.898i$
Analytic cond. $0.854492$
Root an. cond. $0.854492$
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 + (−0.415 + 0.909i)5-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 + (0.841 − 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-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 + ⋯
L(s)  = 1  + (−0.841 − 0.540i)3-s + (−0.415 + 0.909i)5-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 + (0.841 − 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5885930066 - 0.3679276488i\)
\(L(\frac12)\) \(\approx\) \(0.5885930066 - 0.3679276488i\)
\(L(1)\) \(\approx\) \(0.7154237330 - 0.1570080511i\)
\(L(1)\) \(\approx\) \(0.7154237330 - 0.1570080511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
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

−27.67465867345348629902424134448, −26.58329737480015811585319868624, −25.49850538510736261646336201838, −24.49223177811915486572814834846, −23.24389312739810058389658071326, −22.88031351537119879255153084058, −21.671397030812856622789167762683, −20.78318025197260471511965651277, −19.77967216251718773568201242797, −18.71615196449285001223228714293, −17.40695566308492473126558075903, −16.62470716622613758700801680322, −15.85403358827097634603195492335, −15.03133085842618459486009270856, −13.31014814006194099533279388469, −12.32053336217923695082018749942, −11.71085255254585673807308616276, −10.27725532494650186230847372318, −9.41555065996714939859783542182, −8.34177497281486898834897669772, −6.629972601608790080987880261879, −5.814782047160663618348652969951, −4.43675581686091476823149899959, −3.64549594823914281285732006219, −1.331002356556344838002518065777, 0.70817605432551656585729062150, 2.76280579700120862180888961887, 3.948991653743340308148884736477, 5.66816180765879509824725245416, 6.65917394493711670523447004319, 7.27755684575699182144249443934, 8.84755621008033663697997630390, 10.33591184776950267414982617373, 11.15486619453045344214171269893, 11.981044760666855156379874562743, 13.2914236092976831861662073584, 13.99337231736029439080246977041, 15.642622012686964922444701400486, 16.24762159226725324351877770087, 17.44577301183599152388157121497, 18.452961125146385874725375653750, 19.1285521811462715097979253788, 20.047070372203927441940811757838, 21.72544458930173721760019426544, 22.514359932713218621847707843662, 23.06843779752446501613673854111, 24.02778674326688739719117746621, 25.148444598388363451218351977566, 26.13942460620274078301655385294, 27.12044254778922522604310385801

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