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Properties

Label 2-384-48.11-c3-0-12
Degree $2$
Conductor $384$
Sign $-0.134 - 0.990i$
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  + (4.46 + 2.65i)3-s + (5.27 − 5.27i)5-s − 22.9·7-s + (12.9 + 23.7i)9-s + (−10.6 − 10.6i)11-s + (−12.7 + 12.7i)13-s + (37.5 − 9.57i)15-s + 134. i·17-s + (46.9 + 46.9i)19-s + (−102. − 60.8i)21-s + 93.7i·23-s + 69.2i·25-s + (−5.17 + 140. i)27-s + (161. + 161. i)29-s − 120. i·31-s + ⋯
L(s)  = 1  + (0.859 + 0.510i)3-s + (0.472 − 0.472i)5-s − 1.23·7-s + (0.478 + 0.878i)9-s + (−0.291 − 0.291i)11-s + (−0.271 + 0.271i)13-s + (0.646 − 0.164i)15-s + 1.91i·17-s + (0.566 + 0.566i)19-s + (−1.06 − 0.632i)21-s + 0.849i·23-s + 0.554i·25-s + (−0.0369 + 0.999i)27-s + (1.03 + 1.03i)29-s − 0.698i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.134 - 0.990i$
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: \((2,\ 384,\ (\ :3/2),\ -0.134 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.991016363\)
\(L(\frac12)\) \(\approx\) \(1.991016363\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.46 - 2.65i)T \)
good5 \( 1 + (-5.27 + 5.27i)T - 125iT^{2} \)
7 \( 1 + 22.9T + 343T^{2} \)
11 \( 1 + (10.6 + 10.6i)T + 1.33e3iT^{2} \)
13 \( 1 + (12.7 - 12.7i)T - 2.19e3iT^{2} \)
17 \( 1 - 134. iT - 4.91e3T^{2} \)
19 \( 1 + (-46.9 - 46.9i)T + 6.85e3iT^{2} \)
23 \( 1 - 93.7iT - 1.21e4T^{2} \)
29 \( 1 + (-161. - 161. i)T + 2.43e4iT^{2} \)
31 \( 1 + 120. iT - 2.97e4T^{2} \)
37 \( 1 + (2.42 + 2.42i)T + 5.06e4iT^{2} \)
41 \( 1 - 253.T + 6.89e4T^{2} \)
43 \( 1 + (-135. + 135. i)T - 7.95e4iT^{2} \)
47 \( 1 + 468.T + 1.03e5T^{2} \)
53 \( 1 + (-321. + 321. i)T - 1.48e5iT^{2} \)
59 \( 1 + (119. + 119. i)T + 2.05e5iT^{2} \)
61 \( 1 + (310. - 310. i)T - 2.26e5iT^{2} \)
67 \( 1 + (705. + 705. i)T + 3.00e5iT^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + 641. iT - 3.89e5T^{2} \)
79 \( 1 - 1.23e3iT - 4.93e5T^{2} \)
83 \( 1 + (-87.3 + 87.3i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.19e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−10.84519596218161741590886455253, −10.01289592398175539820027473676, −9.407350716696073584414446197748, −8.577664403386105257719445518978, −7.60301314635534231347117719105, −6.31834332128432440698736792428, −5.32691326649495432363098864249, −3.94531894433362182050661922431, −3.08783506218011117165413184332, −1.64006807233212883994374456838, 0.59334020565996074088115725052, 2.61184466128842132903547482451, 2.96552923302956863684904344064, 4.63353232142554592059426698564, 6.18296424026616846128740353305, 6.91339668446254871194040837148, 7.71048909469129626850012207681, 9.003986879553538171315280630702, 9.686154259510259340744660656998, 10.32798118831722638069199240775

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