↑

Properties

Label 1-184-184.5-r1-0-0
Degree $1$
Conductor $184$
Sign $-0.298 + 0.954i$
Analytic cond. $19.7735$
Root an. cond. $19.7735$
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

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

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(19.7735\)
Root analytic conductor: \(19.7735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (1:\ ),\ -0.298 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8568394422 + 1.165621440i\)
\(L(\frac12)\) \(\approx\) \(0.8568394422 + 1.165621440i\)
\(L(1)\) \(\approx\) \(0.9529305323 + 0.3764060990i\)
\(L(1)\) \(\approx\) \(0.9529305323 + 0.3764060990i\)

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

−26.73405252755733353389931454167, −25.549619473118614286106681493180, −24.70779836598888360762022396626, −24.01720160293064106367060322842, −22.884164249471131966846072407200, −22.297529963179995166021016757529, −20.60082248374350301667101024568, −19.924248483266339548630486961912, −18.874005938430866192762758143605, −18.14322203428381424260046143581, −17.28538607821124824660917740932, −15.67781418375296069773084122436, −14.96442099476721083293969264771, −13.892615263446136637314089857705, −12.71283692754921470077593842487, −11.679757139634548791289007346103, −11.24143434869877011718203239578, −9.265522390821020829723199659073, −8.25189657189471432497199309989, −7.394117076632267858224566025391, −6.34105977286475088524176697268, −4.92195563777053325955384859006, −3.368310598157144444984302987548, −2.10056527855310957984367259425, −0.55274647717041969040661494213, 1.35157140745839691594618418791, 3.66150006921416583086493821489, 4.019096227865148964081008792839, 5.32105351607425495117233765417, 6.90548367182453161399064688080, 8.32563225350335569204353311508, 8.89710393189141617537523780551, 10.45403890716272704899127726654, 11.19251206034742275408509219059, 12.10888440270377691044944525580, 13.81577865411083570590821204074, 14.539027861333099608064934352310, 15.62157940668924325316955455181, 16.53987103063876448874001132456, 17.16384236518216348789245246914, 18.78083320732326782700447995922, 19.8382340680908727923082214500, 20.49273729057276553994064065623, 21.45007246579668469751408608135, 22.46268828987077222028556976150, 23.51227735854038088856137718805, 24.21192072884402585403894062175, 25.56403109281313244894920134314, 26.71211356797515207592672270030, 27.14241104665222104609483986704

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