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Properties

Label 2-384-12.11-c3-0-6
Degree $2$
Conductor $384$
Sign $-0.924 - 0.381i$
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  + (4.80 + 1.98i)3-s + 11.9i·5-s − 22.6i·7-s + (19.1 + 19.0i)9-s − 61.9·11-s − 71.3·13-s + (−23.7 + 57.5i)15-s + 74.3i·17-s + 108. i·19-s + (44.9 − 109. i)21-s − 24.7·23-s − 18.6·25-s + (54.2 + 129. i)27-s + 84.1i·29-s − 130. i·31-s + ⋯
L(s)  = 1  + (0.924 + 0.381i)3-s + 1.07i·5-s − 1.22i·7-s + (0.709 + 0.705i)9-s − 1.69·11-s − 1.52·13-s + (−0.408 + 0.990i)15-s + 1.06i·17-s + 1.30i·19-s + (0.467 − 1.13i)21-s − 0.223·23-s − 0.149·25-s + (0.386 + 0.922i)27-s + 0.538i·29-s − 0.755i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.924 - 0.381i$
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: \((2,\ 384,\ (\ :3/2),\ -0.924 - 0.381i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.154152030\)
\(L(\frac12)\) \(\approx\) \(1.154152030\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.80 - 1.98i)T \)
good5 \( 1 - 11.9iT - 125T^{2} \)
7 \( 1 + 22.6iT - 343T^{2} \)
11 \( 1 + 61.9T + 1.33e3T^{2} \)
13 \( 1 + 71.3T + 2.19e3T^{2} \)
17 \( 1 - 74.3iT - 4.91e3T^{2} \)
19 \( 1 - 108. iT - 6.85e3T^{2} \)
23 \( 1 + 24.7T + 1.21e4T^{2} \)
29 \( 1 - 84.1iT - 2.43e4T^{2} \)
31 \( 1 + 130. iT - 2.97e4T^{2} \)
37 \( 1 + 397.T + 5.06e4T^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 + 161. iT - 7.95e4T^{2} \)
47 \( 1 - 72.8T + 1.03e5T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 - 243.T + 2.05e5T^{2} \)
61 \( 1 - 358.T + 2.26e5T^{2} \)
67 \( 1 - 449. iT - 3.00e5T^{2} \)
71 \( 1 - 329.T + 3.57e5T^{2} \)
73 \( 1 + 925.T + 3.89e5T^{2} \)
79 \( 1 + 55.9iT - 4.93e5T^{2} \)
83 \( 1 - 928.T + 5.71e5T^{2} \)
89 \( 1 - 853. iT - 7.04e5T^{2} \)
97 \( 1 + 714.T + 9.12e5T^{2} \)
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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

−10.75186150244858293717249974016, −10.35211213319975982183274034095, −9.897365484100474746302830383404, −8.310095273350699182277368267010, −7.57703868826841207018156625241, −6.99852830766372665222472824770, −5.36868835503093729996025973351, −4.12868813055608643662728013312, −3.14726578574453180998684620967, −2.09316603216440046756264184648, 0.31225409414659223742680683795, 2.22254294448809636063580860442, 2.86049504091982944573727736154, 4.86081279805222983920705770820, 5.26963651861762386821303790272, 6.98767174125398693170397653235, 7.85728957256759668311839161687, 8.744195209309821416051321104203, 9.302322472859348928051970816768, 10.24095467671652940077578735263

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