Properties

Label 1-384-384.299-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: $\chi_{384} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (0:\ ),\ 0.242 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5937047559 - 0.4633317390i\)
\(L(\frac12)\) \(\approx\) \(0.5937047559 - 0.4633317390i\)
\(L(1)\) \(\approx\) \(0.7869015243 - 0.1034538088i\)
\(L(1)\) \(\approx\) \(0.7869015243 - 0.1034538088i\)

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

−24.67901843082913024914384167584, −23.76488234188080461478323579934, −23.10192258303380724479651428485, −21.95550019903781483380980981871, −21.46993564722767776541718736523, −19.969349721523462274488972465375, −19.500793233018441114016146765510, −18.94930977750104070824461016629, −17.44807322567714331506648157767, −16.70744289362307512627810336599, −15.89801829065741435033595376459, −15.06659119434508903997953040622, −14.05904154696375612633393760982, −12.8354039433044418801769774607, −12.16143194237395936320629186439, −11.44322110808644513380489934943, −10.04887965765923935155522279991, −9.15496114522609093344162068742, −8.358671483326760648240980788130, −7.12707788435956479388408606369, −6.273219490385868125370219660082, −4.96260336052111308304082644994, −3.94127939605492129878589944488, −2.95956442729049089833048647985, −1.31549393455941298138828118065, 0.48949155910571284482278771827, 2.45506500183996105444388947757, 3.55356987623735305090441945120, 4.32928950879019261084408254816, 5.87481184280595337497670483732, 6.91316837879726910690345877676, 7.57610732679858674108496997476, 8.794902629180772599231842568961, 10.00182409748251457319784744974, 10.57281562984093692701969897101, 12.11680445484702783718126805243, 12.24363578271911071418500908251, 13.77351611137934844051676150764, 14.59328716459808055842875248562, 15.42607705216420112691882559142, 16.43959906369739700394935156892, 17.096511314428476174969202691566, 18.38720164897654948400871766676, 19.20229444321517914385235939219, 19.81035015398744870824457709976, 20.65259751424893612957896361461, 22.05434027359286547543783256488, 22.69517146247014684608760996949, 23.16233176817388223641030685350, 24.35734417606580056765758640112

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