Properties

Label 1-384-384.35-r0-0-0
Degree $1$
Conductor $384$
Sign $0.970 - 0.242i$
Analytic cond. $1.78328$
Root an. cond. $1.78328$
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  + (0.555 − 0.831i)5-s + (0.923 − 0.382i)7-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (0.382 − 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.195 − 0.980i)29-s + i·31-s + (0.195 − 0.980i)35-s + (−0.831 − 0.555i)37-s + (0.382 − 0.923i)41-s + (−0.980 + 0.195i)43-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)5-s + (0.923 − 0.382i)7-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (0.707 + 0.707i)17-s + (−0.831 + 0.555i)19-s + (0.382 − 0.923i)23-s + (−0.382 − 0.923i)25-s + (0.195 − 0.980i)29-s + i·31-s + (0.195 − 0.980i)35-s + (−0.831 − 0.555i)37-s + (0.382 − 0.923i)41-s + (−0.980 + 0.195i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(1.78328\)
Root analytic conductor: \(1.78328\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.631720711 - 0.2012535543i\)
\(L(\frac12)\) \(\approx\) \(1.631720711 - 0.2012535543i\)
\(L(1)\) \(\approx\) \(1.305060571 - 0.1097795545i\)
\(L(1)\) \(\approx\) \(1.305060571 - 0.1097795545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.555 + 0.831i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.195 - 0.980i)T \)
13 \( 1 + (-0.555 - 0.831i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.831 - 0.555i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (-0.195 + 0.980i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.831 + 0.555i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (0.980 - 0.195i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.195 - 0.980i)T \)
59 \( 1 + (-0.555 + 0.831i)T \)
61 \( 1 + (0.980 + 0.195i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.831 - 0.555i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 + iT \)
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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

−24.66248676788016967729184493186, −23.65515002456457030145918178347, −22.79009972100123768973249681204, −21.73076416497630471651364243644, −21.32739373433954231986660324635, −20.30046795559262373388912444845, −19.0614360757878192775089910563, −18.39290996388386231793508814877, −17.65335845579163401173973789055, −16.7509296016788102670002150824, −15.48555602310084740810265705417, −14.77621129229910881468657930436, −13.88858172735120399500418076050, −13.13938127435804106515465401083, −11.68217357895842635043338942470, −11.06758858081719193877913312972, −10.18317040108989450507472107180, −8.977842266430526919759922716682, −8.09548876458425828799724024167, −6.9895038866615534586891513423, −5.86424369284539072877787555549, −5.15591160405979644554880800088, −3.542268134088534145845870223924, −2.629640490593411818519777417319, −1.30052430006472192509183128720, 1.33641848543762234364350966113, 2.07477001930076423828874800991, 4.023581407606691232190606612926, 4.68361947376115885626985317251, 5.81677037530315083151988284375, 6.91870251494206305119985140067, 8.156460131834255107581106601303, 8.857779288021632153231742474196, 9.99571776244884446903424664452, 10.82695550583977988327315631715, 12.10735666337791881452181520739, 12.70269736818384658164542383086, 13.93458871879559380197086507614, 14.499692968577424454488145352333, 15.65203577452626401594488799934, 16.89169291709083621509054823759, 17.19492949963934934568933356410, 18.2258699075312496098789018386, 19.263356987675120662262810700150, 20.37573170360697020049121792955, 21.00516475116377958011142888264, 21.52609631825570356194394953347, 22.98418439309908634191365546662, 23.60523419581964800135689831540, 24.51580218455127202734494553175

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