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Properties

Label 2-384-12.11-c1-0-4
Degree $2$
Conductor $384$
Sign $0.934 - 0.356i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 0.618i)3-s + 1.23i·5-s − 3.23i·7-s + (2.23 − 2.00i)9-s + 0.763·11-s + 4.47·13-s + (−0.763 − 2.00i)15-s + 6.47i·17-s + 5.23i·19-s + (2.00 + 5.23i)21-s + 6.47·23-s + 3.47·25-s + (−2.38 + 4.61i)27-s − 9.23i·29-s + 0.763i·31-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)3-s + 0.552i·5-s − 1.22i·7-s + (0.745 − 0.666i)9-s + 0.230·11-s + 1.24·13-s + (−0.197 − 0.516i)15-s + 1.56i·17-s + 1.20i·19-s + (0.436 + 1.14i)21-s + 1.34·23-s + 0.694·25-s + (−0.458 + 0.888i)27-s − 1.71i·29-s + 0.137i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07028 + 0.197449i\)
\(L(\frac12)\) \(\approx\) \(1.07028 + 0.197449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 0.618i)T \)
good5 \( 1 - 1.23iT - 5T^{2} \)
7 \( 1 + 3.23iT - 7T^{2} \)
11 \( 1 - 0.763T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 6.47iT - 17T^{2} \)
19 \( 1 - 5.23iT - 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + 9.23iT - 29T^{2} \)
31 \( 1 - 0.763iT - 31T^{2} \)
37 \( 1 + 0.472T + 37T^{2} \)
41 \( 1 + 2.47iT - 41T^{2} \)
43 \( 1 + 2.76iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 - 3.23T + 59T^{2} \)
61 \( 1 + 8.47T + 61T^{2} \)
67 \( 1 + 3.70iT - 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 7.23T + 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 8.47T + 97T^{2} \)
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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

−10.99513743930460302127490586551, −10.70591091554262089597029801492, −9.958742619105365954791435361309, −8.642383833127025757401369485869, −7.44684296800419373091218472860, −6.49958337079848265010822460002, −5.79526940032163107620983650896, −4.25027968734526827701723628527, −3.61708637713807937652164434514, −1.21607159322177550095062969573, 1.11252385058558417766764830712, 2.86233646484382091941252710760, 4.78007082550275247062899098846, 5.36493042952581033648915175603, 6.46217092559972383772333794662, 7.32744672378307609304812102459, 8.884997237430113978137176093514, 9.084718219930451733045288532149, 10.68854169542896677749479178726, 11.39800771162205547392413200490

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