↑

Properties

Label 1-184-184.83-r0-0-0
Degree $1$
Conductor $184$
Sign $-0.298 - 0.954i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 + 0.989i)3-s + (−0.959 + 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (0.415 − 0.909i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (−0.959 + 0.281i)5-s + (−0.654 − 0.755i)7-s + (−0.959 − 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.654 − 0.755i)13-s + (−0.142 − 0.989i)15-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.841 − 0.540i)25-s + (0.415 − 0.909i)27-s + (−0.415 − 0.909i)29-s + (0.142 + 0.989i)31-s + (0.654 − 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.298 - 0.954i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ -0.298 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1550734029 - 0.2109577063i\)
\(L(\frac12)\) \(\approx\) \(0.1550734029 - 0.2109577063i\)
\(L(1)\) \(\approx\) \(0.5822669831 + 0.04810054436i\)
\(L(1)\) \(\approx\) \(0.5822669831 + 0.04810054436i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−28.01201970272462868050200378997, −26.18593913619036164027646374179, −25.76196689258158914965740244260, −24.40525730716724282895709165253, −23.76898184360254975618892550273, −23.00881530895185685150946101628, −21.95932232379582414151998803268, −20.59631761354112878029388725647, −19.537071196887410413652156602889, −18.91699228386683735335496581081, −18.05739443396188256842345893701, −16.81724990196473420161114895590, −15.761085979504354984364867310751, −14.932117419642817550164059325272, −13.32274509845972624575467961525, −12.71447387665450613484968754744, −11.792763046372599037382368970623, −10.82794439519736585329113981913, −9.05775022472202055370144610570, −8.24485246956045211525727307891, −7.09736329907800705256943983174, −6.137322788743229714495686894259, −4.75417513706635832817273765406, −3.19248662776149156085346878907, −1.842456735663416569016603760438, 0.202123593032305903131738024815, 3.08634477274873530652546826232, 3.75318203792433640907499748104, 5.01083677229299172582112567897, 6.33861366014945861220968594928, 7.713125747479423723155252005470, 8.721153471864319316379478232340, 10.19608164664124575429416231663, 10.73523046063263865239608871431, 11.793414231166414705453458459104, 13.1606657951495088748017619664, 14.276744870436135689940801063422, 15.59569572039216586820641872779, 15.93715143662739895840719653938, 16.93687166154823521284184797858, 18.283413453933900410478757395005, 19.3761076450034835668715486184, 20.35100662928330349473325920714, 21.025693980337956936997454020411, 22.46635267118614939847013353951, 22.949243684438220675106153096035, 23.73896764261893053473613979851, 25.28664695175283504391261153716, 26.347562766065496276442265267650, 26.89198853830442485132964931255

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