Properties

Label 2-384-12.11-c3-0-28
Degree $2$
Conductor $384$
Sign $0.810 - 0.585i$
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  + (3.04 + 4.21i)3-s + 9.33i·5-s − 36.3i·7-s + (−8.50 + 25.6i)9-s + 48.4·11-s + 25.8·13-s + (−39.3 + 28.3i)15-s − 74.2i·17-s + 82.9i·19-s + (153. − 110. i)21-s + 179.·23-s + 37.8·25-s + (−133. + 42.1i)27-s − 122. i·29-s + 64.1i·31-s + ⋯
L(s)  = 1  + (0.585 + 0.810i)3-s + 0.834i·5-s − 1.96i·7-s + (−0.314 + 0.949i)9-s + 1.32·11-s + 0.552·13-s + (−0.676 + 0.488i)15-s − 1.05i·17-s + 1.00i·19-s + (1.59 − 1.14i)21-s + 1.62·23-s + 0.303·25-s + (−0.953 + 0.300i)27-s − 0.783i·29-s + 0.371i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.810 - 0.585i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.602568013\)
\(L(\frac12)\) \(\approx\) \(2.602568013\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.04 - 4.21i)T \)
good5 \( 1 - 9.33iT - 125T^{2} \)
7 \( 1 + 36.3iT - 343T^{2} \)
11 \( 1 - 48.4T + 1.33e3T^{2} \)
13 \( 1 - 25.8T + 2.19e3T^{2} \)
17 \( 1 + 74.2iT - 4.91e3T^{2} \)
19 \( 1 - 82.9iT - 6.85e3T^{2} \)
23 \( 1 - 179.T + 1.21e4T^{2} \)
29 \( 1 + 122. iT - 2.43e4T^{2} \)
31 \( 1 - 64.1iT - 2.97e4T^{2} \)
37 \( 1 - 5.01T + 5.06e4T^{2} \)
41 \( 1 - 325. iT - 6.89e4T^{2} \)
43 \( 1 - 321. iT - 7.95e4T^{2} \)
47 \( 1 + 95.9T + 1.03e5T^{2} \)
53 \( 1 + 185. iT - 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 - 198.T + 2.26e5T^{2} \)
67 \( 1 + 23.9iT - 3.00e5T^{2} \)
71 \( 1 - 399.T + 3.57e5T^{2} \)
73 \( 1 - 669.T + 3.89e5T^{2} \)
79 \( 1 - 229. iT - 4.93e5T^{2} \)
83 \( 1 + 321.T + 5.71e5T^{2} \)
89 \( 1 - 131. iT - 7.04e5T^{2} \)
97 \( 1 - 136.T + 9.12e5T^{2} \)
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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

−10.98721869630134487511289290437, −10.04783968760848892789369181303, −9.445991232126793913686793769177, −8.197341231287106156273948372320, −7.20667667678370688486084013795, −6.51473504744041139880510929511, −4.75910668265081495733029884790, −3.84355075367297843327706026967, −3.11940765614824515467016516537, −1.14635855814090701065738052842, 1.12153956736246698794356243912, 2.23875338887436201034434318384, 3.51893176359333953449360416280, 5.09968639841401672694544666736, 6.09494645811704991982923716992, 6.95667896685828897049072767158, 8.504265020346250827381865655647, 8.843698132548598803774632041983, 9.288195637786486309857766284264, 11.13956684567071003027025952691

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