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Properties

Label 1-184-184.77-r0-0-0
Degree $1$
Conductor $184$
Sign $0.448 - 0.893i$
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  + (0.654 − 0.755i)3-s + (0.142 − 0.989i)5-s + (0.415 + 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (−0.959 − 0.281i)25-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (0.142 − 0.989i)5-s + (0.415 + 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (−0.415 + 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (−0.959 − 0.281i)25-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 0.448 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.275420572 - 0.7869074942i\)
\(L(\frac12)\) \(\approx\) \(1.275420572 - 0.7869074942i\)
\(L(1)\) \(\approx\) \(1.263505699 - 0.4576719362i\)
\(L(1)\) \(\approx\) \(1.263505699 - 0.4576719362i\)

Euler product

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

−27.28846489702357397749992690020, −26.52164963878834899736381514418, −25.61279646102160260205383976706, −24.87587432506402448188694227033, −23.36005003076195733847754317966, −22.531181078472283154222029939453, −21.64983743974059978225278112281, −20.672090782811988557525293358252, −19.76352333463751633908024838501, −18.944249638289810117042288220902, −17.49958977692354311107715974952, −16.80684514345235656099747188554, −15.37984632772026336175081562494, −14.50711187571678818912501519533, −14.11947283104738870530831240335, −12.6429044410568062067707794497, −11.02808442379617406689024737984, −10.40497441832858921140089442987, −9.48824152253099412482592539826, −8.04219445510280491609807519489, −7.205338086013071109416193908, −5.71444337689830482126336831411, −4.16658203044831021828228927245, −3.402058582362780747706907932143, −1.93725777085607962335162037592, 1.33017633426803191867488274796, 2.39339015535412656478180691142, 4.03161626710998154349823203562, 5.40652922630202130117991859836, 6.61093614543106629574652835834, 7.911706528099350987891146172871, 8.95788019056997454464353105413, 9.39322204141702259666218406098, 11.56944461474356570851981543178, 12.18935555324315699345040963970, 13.19118489013041100348108948783, 14.26734116208777252800527140029, 15.05248452633136197828278148986, 16.49836192971728590368637909761, 17.34213152592301704389045777940, 18.568947787016445991808094829715, 19.27927185754554764208694172522, 20.305909014754822300714751814758, 21.17343341093151038877081431213, 22.10660979804224333859026892910, 23.76499814056220217315532426842, 24.23147655753079938208369387973, 25.130969733398729997883494153177, 25.73624555383743272648120905164, 27.19912549515798620538257213127

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