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Properties

Label 2-384-12.11-c3-0-15
Degree $2$
Conductor $384$
Sign $0.994 - 0.107i$
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  + (−0.556 − 5.16i)3-s + 10.6i·5-s − 7.90i·7-s + (−26.3 + 5.75i)9-s − 11.7·11-s + 30.5·13-s + (54.8 − 5.90i)15-s + 118. i·17-s − 66.4i·19-s + (−40.8 + 4.40i)21-s + 166.·23-s + 12.4·25-s + (44.4 + 133. i)27-s − 111. i·29-s + 224. i·31-s + ⋯
L(s)  = 1  + (−0.107 − 0.994i)3-s + 0.948i·5-s − 0.426i·7-s + (−0.977 + 0.213i)9-s − 0.322·11-s + 0.652·13-s + (0.943 − 0.101i)15-s + 1.68i·17-s − 0.802i·19-s + (−0.424 + 0.0457i)21-s + 1.50·23-s + 0.0998·25-s + (0.316 + 0.948i)27-s − 0.717i·29-s + 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.994 - 0.107i$
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: \((2,\ 384,\ (\ :3/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.682490419\)
\(L(\frac12)\) \(\approx\) \(1.682490419\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.556 + 5.16i)T \)
good5 \( 1 - 10.6iT - 125T^{2} \)
7 \( 1 + 7.90iT - 343T^{2} \)
11 \( 1 + 11.7T + 1.33e3T^{2} \)
13 \( 1 - 30.5T + 2.19e3T^{2} \)
17 \( 1 - 118. iT - 4.91e3T^{2} \)
19 \( 1 + 66.4iT - 6.85e3T^{2} \)
23 \( 1 - 166.T + 1.21e4T^{2} \)
29 \( 1 + 111. iT - 2.43e4T^{2} \)
31 \( 1 - 224. iT - 2.97e4T^{2} \)
37 \( 1 + 70.6T + 5.06e4T^{2} \)
41 \( 1 - 247. iT - 6.89e4T^{2} \)
43 \( 1 - 98.7iT - 7.95e4T^{2} \)
47 \( 1 - 189.T + 1.03e5T^{2} \)
53 \( 1 + 529. iT - 1.48e5T^{2} \)
59 \( 1 - 811.T + 2.05e5T^{2} \)
61 \( 1 - 833.T + 2.26e5T^{2} \)
67 \( 1 - 2.83iT - 3.00e5T^{2} \)
71 \( 1 - 796.T + 3.57e5T^{2} \)
73 \( 1 + 875.T + 3.89e5T^{2} \)
79 \( 1 + 656. iT - 4.93e5T^{2} \)
83 \( 1 + 237.T + 5.71e5T^{2} \)
89 \( 1 - 868. iT - 7.04e5T^{2} \)
97 \( 1 - 755.T + 9.12e5T^{2} \)
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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

−10.90746821536245493019613516834, −10.39040633898077679564495510395, −8.844843229864734426616597665350, −8.044589762580535678165843489858, −6.93363235112178388113350095635, −6.51446879454661761518564127372, −5.28072888037850276477690702714, −3.62562212490150747512106440541, −2.50788419320440672646265099376, −1.06251116560897570402594862086, 0.73282755587719211496966884912, 2.71221628916112011594792371716, 4.00061979356112389245580588693, 5.11743065434242594491908658085, 5.62539700103097022849938327932, 7.16337332166446029992021531288, 8.548927216130733219658245761967, 9.012939591182058960614459385071, 9.871135305564044439025100771291, 10.91859158716324639953059635926

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