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Properties

Label 2-384-16.11-c6-0-32
Degree $2$
Conductor $384$
Sign $0.684 + 0.728i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 11.0i)3-s + (73.3 + 73.3i)5-s + 291.·7-s + 242. i·9-s + (1.59e3 − 1.59e3i)11-s + (−1.76e3 + 1.76e3i)13-s − 1.61e3i·15-s + 6.75e3·17-s + (−8.59e3 − 8.59e3i)19-s + (−3.21e3 − 3.21e3i)21-s + 1.61e4·23-s − 4.85e3i·25-s + (2.67e3 − 2.67e3i)27-s + (7.77e3 − 7.77e3i)29-s + 1.84e4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.587 + 0.587i)5-s + 0.849·7-s + 0.333i·9-s + (1.19 − 1.19i)11-s + (−0.804 + 0.804i)13-s − 0.479i·15-s + 1.37·17-s + (−1.25 − 1.25i)19-s + (−0.346 − 0.346i)21-s + 1.32·23-s − 0.310i·25-s + (0.136 − 0.136i)27-s + (0.318 − 0.318i)29-s + 0.618i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.529629630\)
\(L(\frac12)\) \(\approx\) \(2.529629630\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (-73.3 - 73.3i)T + 1.56e4iT^{2} \)
7 \( 1 - 291.T + 1.17e5T^{2} \)
11 \( 1 + (-1.59e3 + 1.59e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (1.76e3 - 1.76e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 6.75e3T + 2.41e7T^{2} \)
19 \( 1 + (8.59e3 + 8.59e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.61e4T + 1.48e8T^{2} \)
29 \( 1 + (-7.77e3 + 7.77e3i)T - 5.94e8iT^{2} \)
31 \( 1 - 1.84e4iT - 8.87e8T^{2} \)
37 \( 1 + (5.21e4 + 5.21e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 3.93e4iT - 4.75e9T^{2} \)
43 \( 1 + (4.81e3 - 4.81e3i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.11e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.03e5 - 1.03e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.52e5 - 1.52e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-7.55e4 + 7.55e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (-3.37e5 - 3.37e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 3.06e5T + 1.28e11T^{2} \)
73 \( 1 + 1.66e5iT - 1.51e11T^{2} \)
79 \( 1 + 1.31e5iT - 2.43e11T^{2} \)
83 \( 1 + (3.84e5 + 3.84e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 3.79e5iT - 4.96e11T^{2} \)
97 \( 1 + 5.77e5T + 8.32e11T^{2} \)
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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

−10.42018904178564554482997937939, −9.217145345908156593836172568249, −8.435856310155244762196804073712, −7.13150299859869243763649039791, −6.51896625068422710248152460708, −5.50028444021620082654466659149, −4.42741273366947835084003216126, −2.95372764655132374948841115419, −1.77231484745656898999768191760, −0.69847611349870369175602610588, 1.05265048254410203961324606701, 1.91660153553200465990637206436, 3.64644324448519443404628578175, 4.83612463410433894467650697678, 5.35891834836938147611331077249, 6.58723191878857213749547206779, 7.69745573104503369679826607064, 8.716879290641847743628457415646, 9.738798860592815642491772054774, 10.23140319907866091201288656755

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