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Properties

Label 1-384-384.221-r1-0-0
Degree $1$
Conductor $384$
Sign $-0.242 - 0.970i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
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

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

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ -0.242 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9508285904 - 1.218374241i\)
\(L(\frac12)\) \(\approx\) \(0.9508285904 - 1.218374241i\)
\(L(1)\) \(\approx\) \(1.024142809 - 0.2671893268i\)
\(L(1)\) \(\approx\) \(1.024142809 - 0.2671893268i\)

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

−24.81903502824464755569179820425, −23.345786093139664301818952886447, −22.72834874619336768885117557067, −22.263371252415314335297040514657, −20.93494014194583283401152693970, −20.28927741382613825188175922128, −19.25674566128308317133931122009, −18.21615733947257990634618099321, −17.81191768668272527170004592327, −16.49228429532774826504754851095, −15.78550209307039265429373777376, −14.64996809983848025897494662074, −13.92494323898629587111048231334, −12.930265925467231902487110778836, −12.11933593744798967203577021301, −10.67914399720015951303839554312, −10.13077399297147377060000834237, −9.3399026009828728342531990177, −7.83288164929222883825073110433, −6.96099277407708659468882568634, −6.10812296379758167109478553935, −5.002509371999251687536832359984, −3.46202566937414804281088960338, −2.78532014021759370625048065059, −1.25853939358796373319301621106, 0.44847713838011301716975848647, 1.745334131463798079998272228264, 3.10968302470041835959365619967, 4.19540821329243667570218379188, 5.73388651648750005080454973735, 5.989930823219670912176709360689, 7.50728772479253174369911642321, 8.74365028715098216697704766904, 9.31863462437570939957560264364, 10.289456555161660987213059686957, 11.5726367715663806781509008048, 12.39942031997748656837518913805, 13.50104279141417295135917786266, 13.82904493604519311065786340747, 15.41957452858903058385812219844, 16.188028968153214133466824918396, 16.81521160597358759449604578136, 17.857288409880366506506282703083, 18.9670983024332882906848361082, 19.50520197975624927732207955917, 20.73971519998907280587022949539, 21.4266403511201373127967809898, 22.12432986684980810327139193547, 23.2821808313330915979132610368, 24.15981815838492582418246224330

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