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Properties

Label 1-184-184.19-r0-0-0
Degree $1$
Conductor $184$
Sign $0.987 + 0.155i$
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

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

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 0.987 + 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534519674 + 0.1199996324i\)
\(L(\frac12)\) \(\approx\) \(1.534519674 + 0.1199996324i\)
\(L(1)\) \(\approx\) \(1.375872903 + 0.02920764761i\)
\(L(1)\) \(\approx\) \(1.375872903 + 0.02920764761i\)

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

−27.23150090631678851996629979621, −25.97119401196927050178649099136, −25.57635526038649846980626057138, −24.510448540146354506846280721112, −23.538937544140244354777856753067, −22.0599440560117452586665271492, −21.51903025574205405049247123261, −20.29527584041487400854634697762, −19.79668920657619280541287745901, −18.76897826974266060511538005435, −17.23120167737633076975714096712, −16.32172062830165948237453137618, −15.67932333311386245865842834126, −14.30207768752383254626376174116, −13.3747315427669736378535909114, −12.73903041777271818712740968591, −11.0303857775448606198659683625, −9.93388604852230549732492952322, −8.96934343226237932334479452187, −8.3596252835337409226488670943, −6.68553459784446544452029274756, −5.44080493682955045699705609982, −4.031735375886905429700782118420, −3.14732917749582061455839719537, −1.40048135390587814936316806411, 1.737607935889341922867951175312, 2.92081538695256425812557698247, 3.88378584335253963900971639370, 6.0542911130037704159338611736, 6.760444337948418036780871207777, 7.83494982663413497357478174927, 9.34996383756056159819953624753, 9.801465137336407832964804030249, 11.411453113905384280920925935241, 12.575217832389256086658985144231, 13.56231909783601654054002192835, 14.38808501995440093723781688241, 15.284970101614994813755500639477, 16.449327721642189333272278393798, 17.977892968847007965629074306615, 18.56712196538657401498337725980, 19.424324143662187162632216076814, 20.42909322772034478147347851676, 21.43959704478527840524497749956, 22.692291525173923030305466695803, 23.19846275195854143705692161537, 24.87542321076526017999227640104, 25.327088974329058287497548012262, 26.102910902378126046116627396, 26.95256142126164344207024353835

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