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Properties

Label 1-384-384.323-r0-0-0
Degree $1$
Conductor $384$
Sign $0.242 + 0.970i$
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  + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.980 + 0.195i)11-s + (0.831 − 0.555i)13-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.980 − 0.195i)29-s + i·31-s + (−0.980 − 0.195i)35-s + (0.555 − 0.831i)37-s + (−0.382 + 0.923i)41-s + (−0.195 − 0.980i)43-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.980 + 0.195i)11-s + (0.831 − 0.555i)13-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.980 − 0.195i)29-s + i·31-s + (−0.980 − 0.195i)35-s + (0.555 − 0.831i)37-s + (−0.382 + 0.923i)41-s + (−0.195 − 0.980i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(1.78328\)
Root analytic conductor: \(1.78328\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ 0.242 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9685689531 + 0.7558786299i\)
\(L(\frac12)\) \(\approx\) \(0.9685689531 + 0.7558786299i\)
\(L(1)\) \(\approx\) \(1.032605980 + 0.2900257275i\)
\(L(1)\) \(\approx\) \(1.032605980 + 0.2900257275i\)

Euler product

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

−24.24780200277349561204384593510, −23.57013230448024525183402675853, −22.53180046268566059446929385599, −21.739688397577931399468168575119, −20.62135293431009955521553893994, −20.34431034198369627018487597008, −18.84569186686539255970470447585, −18.35504166583123200200278172842, −17.14831848591658460389184519978, −16.33609873983899078760686489224, −15.80086082493738479437425175680, −14.33321724648601332672181326863, −13.36038449958735042173338651566, −13.05693027150609826297823768132, −11.77448069428046438846507551750, −10.59983841417693175333980139418, −9.73051908481954809731722859185, −9.003847664751273888412100064831, −7.80335921424966760164698224137, −6.62009054008449808729360274354, −5.74263679735265640738078116790, −4.73372698161222297794640576194, −3.38918569876842803204686504753, −2.274280874860921272426499080783, −0.74903931857125667414270120752, 1.59039237463270749006156770469, 2.861905814507919674770635509405, 3.65187439496210680577907872936, 5.610562989637723977004306240309, 5.83104021492067801774148384544, 7.151162652982257074184963995613, 8.18567800521159083815026734947, 9.47030937909651664747652350805, 10.14126964445501143196079405120, 10.94479523822632295958072351459, 12.34995142255482431531312689284, 13.109054444814872502664586221213, 13.89012397811910614414238974129, 15.00988156355296372814313861192, 15.81884780020144111909490492770, 16.721826359833941267687022738963, 17.905591773354833281274560422847, 18.45290251840655236892114497356, 19.28847761806948531752451114715, 20.48933181629388779176719640404, 21.27548745580467764178176576024, 22.08782889857553394086746218210, 22.93680300939117933977088515145, 23.627986982809625178782304183893, 25.03912919232998021993512444634

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