Properties

Label 1-184-184.13-r0-0-0
Degree $1$
Conductor $184$
Sign $0.904 + 0.426i$
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  + (−0.415 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096430311 + 0.2452840995i\)
\(L(\frac12)\) \(\approx\) \(1.096430311 + 0.2452840995i\)
\(L(1)\) \(\approx\) \(1.053181742 + 0.05010316494i\)
\(L(1)\) \(\approx\) \(1.053181742 + 0.05010316494i\)

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

−26.915381726302442126593446757462, −26.76332677562229340532085695702, −25.10914783227855068236599481910, −24.33790240975948910387123956270, −23.40560516455110084136190683947, −22.01041326323842881357217919618, −21.604441124815172699409289248146, −20.45081748855110972918755593365, −19.92216430977228654457442516681, −18.0186668670721302368569371083, −17.354223123692094383550161091040, −16.53919338758244049176675972546, −15.59840976130160789747795768919, −14.330898520257014504123198015634, −13.530851218133138040376713117596, −12.08217387459161564969708675038, −11.108497716005468028743908654872, −10.12844792573466599721365098032, −9.122779032863095780876379889969, −8.10850009707166780948558801147, −6.41325594748975876207419712711, −5.1349172293090428292856580802, −4.607279747143575812112713290001, −2.973756229563158251888464118684, −1.02410496499084995634330866299, 1.78785537085376782279660421714, 2.48350064291377615833644361875, 4.66472773968136928719284814212, 5.79971507199006960389576976159, 6.87920736614847368015043543082, 7.70711744821147734281110394212, 9.154975238813320613162237650952, 10.40118762990124445637343669553, 11.558324937704692986814770339065, 12.26413802303076925042363716836, 13.5752671160095973531149338599, 14.39229825339428872774367564674, 15.34702512367836620994045330197, 17.060470913851118498590026218011, 17.71826378047358254228230147190, 18.37395761987758425244392862294, 19.36833143665657656535764234111, 20.55471469972759486774290445102, 21.87650092345576273853374536608, 22.40788680615405090742887368926, 23.55918533817111922762434281564, 24.59059645668261220472864246148, 25.13157403918846640444132211929, 26.26037868240473171954751322522, 27.31918655029456034285205334947

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