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Properties

Label 2-384-16.11-c6-0-37
Degree $2$
Conductor $384$
Sign $0.866 - 0.499i$
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 + (95.7 + 95.7i)5-s + 338.·7-s + 242. i·9-s + (1.59e3 − 1.59e3i)11-s + (−602. + 602. i)13-s + 2.11e3i·15-s − 1.41e3·17-s + (6.64e3 + 6.64e3i)19-s + (3.73e3 + 3.73e3i)21-s + 1.49e4·23-s + 2.71e3i·25-s + (−2.67e3 + 2.67e3i)27-s + (2.56e4 − 2.56e4i)29-s − 5.22e4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.766 + 0.766i)5-s + 0.987·7-s + 0.333i·9-s + (1.19 − 1.19i)11-s + (−0.274 + 0.274i)13-s + 0.625i·15-s − 0.287·17-s + (0.968 + 0.968i)19-s + (0.403 + 0.403i)21-s + 1.23·23-s + 0.173i·25-s + (−0.136 + 0.136i)27-s + (1.05 − 1.05i)29-s − 1.75i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.866 - 0.499i$
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: \((2,\ 384,\ (\ :3),\ 0.866 - 0.499i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.034768146\)
\(L(\frac12)\) \(\approx\) \(4.034768146\)
\(L(4)\) 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 \( 1 + (-95.7 - 95.7i)T + 1.56e4iT^{2} \)
7 \( 1 - 338.T + 1.17e5T^{2} \)
11 \( 1 + (-1.59e3 + 1.59e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (602. - 602. i)T - 4.82e6iT^{2} \)
17 \( 1 + 1.41e3T + 2.41e7T^{2} \)
19 \( 1 + (-6.64e3 - 6.64e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.49e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.56e4 + 2.56e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 5.22e4iT - 8.87e8T^{2} \)
37 \( 1 + (-1.12e4 - 1.12e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 7.80e4iT - 4.75e9T^{2} \)
43 \( 1 + (6.29e4 - 6.29e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 5.91e4iT - 1.07e10T^{2} \)
53 \( 1 + (-4.56e4 - 4.56e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (4.01e4 - 4.01e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-6.80e4 + 6.80e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (7.58e4 + 7.58e4i)T + 9.04e10iT^{2} \)
71 \( 1 + 5.20e5T + 1.28e11T^{2} \)
73 \( 1 - 3.49e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.84e5iT - 2.43e11T^{2} \)
83 \( 1 + (-7.55e4 - 7.55e4i)T + 3.26e11iT^{2} \)
89 \( 1 - 8.81e5iT - 4.96e11T^{2} \)
97 \( 1 - 4.63e5T + 8.32e11T^{2} \)
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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

−10.34340180120315192346596011505, −9.521940522388374184856499087729, −8.657310778917953749020749351146, −7.72215578067676135566140855481, −6.52156412815437558066664628390, −5.68202174237897552157371520564, −4.43739194410773653473188986229, −3.32985782422340950360178637395, −2.22520269187569974823893647040, −1.04456682671223267724981689854, 1.16187868773991400364102893383, 1.61016591182342170298186601914, 2.99253094496348458149487265006, 4.67293246024261359128017768627, 5.13141305798869897618676795587, 6.69186771448655051921664060896, 7.35097758732270338034699808361, 8.673632876349769261788733807379, 9.119067617352168743243526414373, 10.06027568798016923584786786610

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