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Properties

Label 2-384-16.11-c6-0-47
Degree $2$
Conductor $384$
Sign $-0.636 - 0.771i$
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 + (−107. − 107. i)5-s − 5.36·7-s + 242. i·9-s + (−48.9 + 48.9i)11-s + (2.11e3 − 2.11e3i)13-s + 2.37e3i·15-s − 3.88e3·17-s + (−5.87e3 − 5.87e3i)19-s + (59.1 + 59.1i)21-s + 3.23e3·23-s + 7.63e3i·25-s + (2.67e3 − 2.67e3i)27-s + (3.05e4 − 3.05e4i)29-s − 5.40e4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.862 − 0.862i)5-s − 0.0156·7-s + 0.333i·9-s + (−0.0367 + 0.0367i)11-s + (0.964 − 0.964i)13-s + 0.704i·15-s − 0.790·17-s + (−0.856 − 0.856i)19-s + (0.00639 + 0.00639i)21-s + 0.266·23-s + 0.488i·25-s + (0.136 − 0.136i)27-s + (1.25 − 1.25i)29-s − 1.81i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.636 - 0.771i$
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: \((2,\ 384,\ (\ :3),\ -0.636 - 0.771i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.6078108403\)
\(L(\frac12)\) \(\approx\) \(0.6078108403\)
\(L(4)\) 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 \( 1 + (107. + 107. i)T + 1.56e4iT^{2} \)
7 \( 1 + 5.36T + 1.17e5T^{2} \)
11 \( 1 + (48.9 - 48.9i)T - 1.77e6iT^{2} \)
13 \( 1 + (-2.11e3 + 2.11e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 3.88e3T + 2.41e7T^{2} \)
19 \( 1 + (5.87e3 + 5.87e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 3.23e3T + 1.48e8T^{2} \)
29 \( 1 + (-3.05e4 + 3.05e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 5.40e4iT - 8.87e8T^{2} \)
37 \( 1 + (3.36e4 + 3.36e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 2.90e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.45e4 - 2.45e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 5.81e4iT - 1.07e10T^{2} \)
53 \( 1 + (2.70e4 + 2.70e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (-2.25e5 + 2.25e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.78e3 + 1.78e3i)T - 5.15e10iT^{2} \)
67 \( 1 + (-1.22e5 - 1.22e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.13e5T + 1.28e11T^{2} \)
73 \( 1 - 3.37e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.15e5iT - 2.43e11T^{2} \)
83 \( 1 + (-3.38e5 - 3.38e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 5.27e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.55e6T + 8.32e11T^{2} \)
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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

−9.731530677709502139287900387684, −8.447338605688169507968017451508, −8.149559225342642598247573145611, −6.87633986514608926923583761566, −5.91777536956397050596766385042, −4.77178812748988926357797319695, −3.92807851424006251852360856851, −2.39198294870511121035901782277, −0.847852787636186190648165775075, −0.19355444474476727355425148675, 1.50338619119115750951951363543, 3.13912536226643110946030489402, 3.96166908883537323711145713149, 4.98762467195107189514699102987, 6.50392000090568922903833899588, 6.84098519547803863905960887056, 8.276015371891409092675637889663, 8.969526430435043172493792180660, 10.39487571373891026304836461724, 10.81407824653237849528913577134

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