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Properties

Label 2-384-48.11-c3-0-17
Degree $2$
Conductor $384$
Sign $0.200 + 0.979i$
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  + (−3.47 − 3.86i)3-s + (−13.5 + 13.5i)5-s − 19.7·7-s + (−2.80 + 26.8i)9-s + (20.5 + 20.5i)11-s + (−36.7 + 36.7i)13-s + (99.6 + 5.19i)15-s + 4.20i·17-s + (−38.6 − 38.6i)19-s + (68.6 + 76.1i)21-s − 69.4i·23-s − 243. i·25-s + (113. − 82.5i)27-s + (−23.0 − 23.0i)29-s + 219. i·31-s + ⋯
L(s)  = 1  + (−0.669 − 0.742i)3-s + (−1.21 + 1.21i)5-s − 1.06·7-s + (−0.103 + 0.994i)9-s + (0.562 + 0.562i)11-s + (−0.783 + 0.783i)13-s + (1.71 + 0.0893i)15-s + 0.0599i·17-s + (−0.466 − 0.466i)19-s + (0.713 + 0.791i)21-s − 0.629i·23-s − 1.95i·25-s + (0.808 − 0.588i)27-s + (−0.147 − 0.147i)29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.200 + 0.979i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.200 + 0.979i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3155352478\)
\(L(\frac12)\) \(\approx\) \(0.3155352478\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.47 + 3.86i)T \)
good5 \( 1 + (13.5 - 13.5i)T - 125iT^{2} \)
7 \( 1 + 19.7T + 343T^{2} \)
11 \( 1 + (-20.5 - 20.5i)T + 1.33e3iT^{2} \)
13 \( 1 + (36.7 - 36.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 4.20iT - 4.91e3T^{2} \)
19 \( 1 + (38.6 + 38.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 69.4iT - 1.21e4T^{2} \)
29 \( 1 + (23.0 + 23.0i)T + 2.43e4iT^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 + (68.0 + 68.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 325.T + 6.89e4T^{2} \)
43 \( 1 + (-36.9 + 36.9i)T - 7.95e4iT^{2} \)
47 \( 1 - 192.T + 1.03e5T^{2} \)
53 \( 1 + (-461. + 461. i)T - 1.48e5iT^{2} \)
59 \( 1 + (0.977 + 0.977i)T + 2.05e5iT^{2} \)
61 \( 1 + (216. - 216. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-27.8 - 27.8i)T + 3.00e5iT^{2} \)
71 \( 1 + 786. iT - 3.57e5T^{2} \)
73 \( 1 - 510. iT - 3.89e5T^{2} \)
79 \( 1 + 230. iT - 4.93e5T^{2} \)
83 \( 1 + (-593. + 593. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 805.T + 9.12e5T^{2} \)
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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

−10.80163907024173593972538013278, −10.10631896640436402653239360304, −8.771761087827934552592516188735, −7.47221906755566794410094998438, −6.86483563946570483757732188767, −6.41974088980838823494104337680, −4.72791198014877515691139155532, −3.53093465147010552607152358568, −2.28277430609239665219503369134, −0.18839787270306812203313683566, 0.71875554294141869922569531122, 3.36496647844652003355948126985, 4.12393978659000964447115610205, 5.18098028691359882382743323864, 6.12668545544975562490942998322, 7.41012328704936221610994179239, 8.511412820620035024495416552354, 9.329104034278996705568676115783, 10.13346390020878829532972429097, 11.24703296606262239119508727876

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