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Properties

Label 2-384-24.11-c3-0-41
Degree $2$
Conductor $384$
Sign $0.290 + 0.956i$
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  + (−2.44 + 4.58i)3-s + 17.2·5-s + 0.269i·7-s + (−15 − 22.4i)9-s − 41.1i·11-s − 64.6i·13-s + (−42.3 + 79.1i)15-s − 63.4i·17-s − 154.·19-s + (−1.23 − 0.660i)21-s − 43.3·23-s + 173.·25-s + (139. − 13.7i)27-s − 240.·29-s − 79.1i·31-s + ⋯
L(s)  = 1  + (−0.471 + 0.881i)3-s + 1.54·5-s + 0.0145i·7-s + (−0.555 − 0.831i)9-s − 1.12i·11-s − 1.37i·13-s + (−0.728 + 1.36i)15-s − 0.904i·17-s − 1.85·19-s + (−0.0128 − 0.00686i)21-s − 0.392·23-s + 1.38·25-s + (0.995 − 0.0979i)27-s − 1.54·29-s − 0.458i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.290 + 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.396823894\)
\(L(\frac12)\) \(\approx\) \(1.396823894\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.44 - 4.58i)T \)
good5 \( 1 - 17.2T + 125T^{2} \)
7 \( 1 - 0.269iT - 343T^{2} \)
11 \( 1 + 41.1iT - 1.33e3T^{2} \)
13 \( 1 + 64.6iT - 2.19e3T^{2} \)
17 \( 1 + 63.4iT - 4.91e3T^{2} \)
19 \( 1 + 154.T + 6.85e3T^{2} \)
23 \( 1 + 43.3T + 1.21e4T^{2} \)
29 \( 1 + 240.T + 2.43e4T^{2} \)
31 \( 1 + 79.1iT - 2.97e4T^{2} \)
37 \( 1 + 132. iT - 5.06e4T^{2} \)
41 \( 1 - 101. iT - 6.89e4T^{2} \)
43 \( 1 + 120.T + 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 + 6.17T + 1.48e5T^{2} \)
59 \( 1 - 45.8iT - 2.05e5T^{2} \)
61 \( 1 + 651. iT - 2.26e5T^{2} \)
67 \( 1 - 685.T + 3.00e5T^{2} \)
71 \( 1 - 836.T + 3.57e5T^{2} \)
73 \( 1 - 285.T + 3.89e5T^{2} \)
79 \( 1 - 940. iT - 4.93e5T^{2} \)
83 \( 1 + 1.01e3iT - 5.71e5T^{2} \)
89 \( 1 - 432. iT - 7.04e5T^{2} \)
97 \( 1 - 1.00e3T + 9.12e5T^{2} \)
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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

−10.68009259695732068257005460181, −9.874545304145916037422290852752, −9.158027954767424173518306501912, −8.190879975629750848968788788580, −6.47683872604626462976268392128, −5.77569786687684336851245086385, −5.12217967308100346998987715580, −3.61796578053825311822453518478, −2.35230738961978979277341729680, −0.46346802451615248199005524492, 1.81493841636855457344196936608, 2.05047045044042402139240085462, 4.33602897597073461059352141375, 5.53422408700776010234816110549, 6.43538327661158666667380333772, 6.96280346373057083805137349733, 8.356060667871856292823548979528, 9.329103568223802500265948439320, 10.23001093117156583829397301393, 11.04528088758458714245586902345

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