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Properties

Label 1-184-184.109-r1-0-0
Degree $1$
Conductor $184$
Sign $0.987 - 0.155i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(19.7735\)
Root analytic conductor: \(19.7735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (1:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661146498 - 0.1299018661i\)
\(L(\frac12)\) \(\approx\) \(1.661146498 - 0.1299018661i\)
\(L(1)\) \(\approx\) \(1.045678343 - 0.1422024043i\)
\(L(1)\) \(\approx\) \(1.045678343 - 0.1422024043i\)

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

−26.75060993620521975785957339776, −26.56694691665266034845477400662, −25.14422717529505432291112954680, −23.86292763750527843811652000758, −23.219135828764171622525810906687, −22.195786886277186130894183168384, −21.25755467486273045791769244400, −20.78113658585207500435132529815, −19.01091788227030132272922236888, −18.04278329826471417189335310528, −17.52549691436486314765628908621, −16.22628754636623766619991537613, −15.40640507603877760730802344807, −14.2179738400308502309218838076, −13.37136319882446844745703170503, −11.60891676203466849885308551558, −11.06833908995669660671222543466, −10.27179952330984261115203235310, −8.97314556004075884540316890098, −7.498965845700871241031741477948, −6.321181057037255322795358623864, −5.35369912253302496297731130194, −4.150586060103357411537533255537, −2.69190724033924300121551517947, −0.83345858537517101493355736984, 1.10894181401977350659614487255, 2.01385351334198306721658794380, 4.34239432868845738431912417281, 5.35134050401507171877333581366, 6.14897678109512839258255819311, 7.76762257676600188955024741399, 8.4911942020053748349630841619, 10.09518919703495182095346042287, 11.04491808844431737009124199059, 12.27623510127824622255748459038, 12.84223810517780593335533117786, 13.96713075932437160776070445678, 15.36424215287487693701348190584, 16.437598493941662719794080110570, 17.42826000968286004852705243471, 17.9991538522005113077003872534, 19.01106870737472583439701955983, 20.54268700643616049329111741886, 21.079285166701065363184854872272, 22.23391419384275388562737157789, 23.47844648064579424620983035179, 23.932237189312673427418839311368, 24.98209618125141159610275554708, 25.70516273331231641254107937725, 27.39563339040711765841278787941

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