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Properties

Label 2-384-8.5-c3-0-14
Degree $2$
Conductor $384$
Sign $0.707 + 0.707i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 10.4i·5-s − 6.42·7-s − 9·9-s − 61.6i·11-s − 64.8i·13-s − 31.2·15-s − 75.6·17-s − 10.3i·19-s − 19.2i·21-s + 156.·23-s + 16.3·25-s − 27i·27-s + 53.7i·29-s + 227.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.932i·5-s − 0.346·7-s − 0.333·9-s − 1.69i·11-s − 1.38i·13-s − 0.538·15-s − 1.07·17-s − 0.124i·19-s − 0.200i·21-s + 1.42·23-s + 0.131·25-s − 0.192i·27-s + 0.344i·29-s + 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.338461265\)
\(L(\frac12)\) \(\approx\) \(1.338461265\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
good5 \( 1 - 10.4iT - 125T^{2} \)
7 \( 1 + 6.42T + 343T^{2} \)
11 \( 1 + 61.6iT - 1.33e3T^{2} \)
13 \( 1 + 64.8iT - 2.19e3T^{2} \)
17 \( 1 + 75.6T + 4.91e3T^{2} \)
19 \( 1 + 10.3iT - 6.85e3T^{2} \)
23 \( 1 - 156.T + 1.21e4T^{2} \)
29 \( 1 - 53.7iT - 2.43e4T^{2} \)
31 \( 1 - 227.T + 2.97e4T^{2} \)
37 \( 1 - 10.3iT - 5.06e4T^{2} \)
41 \( 1 + 70.4T + 6.89e4T^{2} \)
43 \( 1 + 298. iT - 7.95e4T^{2} \)
47 \( 1 + 89.9T + 1.03e5T^{2} \)
53 \( 1 - 388. iT - 1.48e5T^{2} \)
59 \( 1 + 324iT - 2.05e5T^{2} \)
61 \( 1 + 324iT - 2.26e5T^{2} \)
67 \( 1 + 920. iT - 3.00e5T^{2} \)
71 \( 1 - 995.T + 3.57e5T^{2} \)
73 \( 1 - 362.T + 3.89e5T^{2} \)
79 \( 1 + 1.09e3T + 4.93e5T^{2} \)
83 \( 1 + 791. iT - 5.71e5T^{2} \)
89 \( 1 + 150.T + 7.04e5T^{2} \)
97 \( 1 + 1.87e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94990260948332287058950926757, −10.08207012006162553646364478139, −8.930774948501923010197599156662, −8.178360241452430017176344057191, −6.85387930554797081692058598828, −6.03312912555073074319391660476, −4.94694672132284673435557998511, −3.36036134816789442073500953724, −2.88471657120935676578383145642, −0.49110437182719285386602376106, 1.27693681370273272238060629301, 2.42482035271289104433297996357, 4.30823829320853091540522653307, 4.96531360452056762308303483580, 6.56490473534773537384508999478, 7.05777120539566782326530413174, 8.342761985885729588086364141534, 9.184365952362803135646726775166, 9.872560341839893490451069111918, 11.26968821674925452091635688460

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