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Properties

Label 1-184-184.53-r1-0-0
Degree $1$
Conductor $184$
Sign $0.257 - 0.966i$
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  + (−0.415 + 0.909i)3-s + (−0.654 + 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.841 − 0.540i)13-s + (−0.415 − 0.909i)15-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (0.959 + 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)3-s + (−0.654 + 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.841 − 0.540i)13-s + (−0.415 − 0.909i)15-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (0.959 + 0.281i)29-s + (0.415 + 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.257 - 0.966i$
Analytic conductor: \(19.7735\)
Root analytic conductor: \(19.7735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (1:\ ),\ 0.257 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003808807700 + 0.002926233417i\)
\(L(\frac12)\) \(\approx\) \(0.003808807700 + 0.002926233417i\)
\(L(1)\) \(\approx\) \(0.5190468288 + 0.2984292987i\)
\(L(1)\) \(\approx\) \(0.5190468288 + 0.2984292987i\)

Euler product

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

−27.30824355904281210785736190414, −26.21134685859200983884375313162, −25.09498557233903957650442084831, −24.14432574266773556853948961160, −23.54762487797462137605017992298, −22.710859668396462271667753553063, −21.50594276269671232823343456459, −20.209731732634361751079418468945, −19.18263022528681526479238689462, −18.938249984936521524310509968375, −17.13255776771438252985280948606, −16.74948326063341926361452300939, −15.724119320960808083269512587614, −14.11276532190767954504454606352, −13.218411319996888789631528719109, −12.32328708146612973193006614300, −11.541801334827341563100100553335, −10.24179918406652235552485080526, −8.80597146772457721503950072239, −7.77816291956335361140222560341, −6.79423464994931076413795090207, −5.63921418898472549272468678255, −4.30682043182885328937908812443, −2.812375170837923656564810397732, −0.99068735657159005038377167472, 0.00219852629933073348943925342, 2.65877388323446256876640973316, 3.675827778867485876707647576317, 4.9207557278432289963611845706, 6.16300577152283257749272544007, 7.273574535549272405521988659460, 8.69575477068908738184821503035, 10.09270337218359017874289408982, 10.4190400978393498229294421953, 12.00816315669332625213721577618, 12.47407006109000677596943316370, 14.427991548014498640472501075448, 15.17631090829732162383689749523, 15.84129852771986831726177775022, 16.97045169277468784355829989061, 18.00143666485051214250145258814, 19.18662713623474829388327002196, 19.97427028698560955661153072110, 21.26117707585048237885554282508, 22.1240262271571158989781880654, 22.927754531831693178227623083766, 23.45363130745522990603471648858, 25.25049412900066640425598018331, 25.89216450911192355854138094521, 26.937230843943740251480437082507

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