Properties

Label 2-384-48.5-c2-0-7
Degree $2$
Conductor $384$
Sign $-0.0677 - 0.997i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 + 0.737i)3-s + (−1.57 + 1.57i)5-s − 3.64i·7-s + (7.91 − 4.29i)9-s + (−1.19 + 1.19i)11-s + (14.6 − 14.6i)13-s + (3.41 − 5.74i)15-s + 28.0i·17-s + (−12.5 + 12.5i)19-s + (2.69 + 10.6i)21-s − 29.2·23-s + 20.0i·25-s + (−19.8 + 18.3i)27-s + (19.3 + 19.3i)29-s + 11.6·31-s + ⋯
L(s)  = 1  + (−0.969 + 0.245i)3-s + (−0.314 + 0.314i)5-s − 0.520i·7-s + (0.878 − 0.476i)9-s + (−0.108 + 0.108i)11-s + (1.12 − 1.12i)13-s + (0.227 − 0.382i)15-s + 1.65i·17-s + (−0.662 + 0.662i)19-s + (0.128 + 0.504i)21-s − 1.27·23-s + 0.801i·25-s + (−0.734 + 0.678i)27-s + (0.667 + 0.667i)29-s + 0.375·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0677 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.0677 - 0.997i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.0677 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.582404 + 0.623319i\)
\(L(\frac12)\) \(\approx\) \(0.582404 + 0.623319i\)
\(L(2)\) 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 \( 1 + (2.90 - 0.737i)T \)
good5 \( 1 + (1.57 - 1.57i)T - 25iT^{2} \)
7 \( 1 + 3.64iT - 49T^{2} \)
11 \( 1 + (1.19 - 1.19i)T - 121iT^{2} \)
13 \( 1 + (-14.6 + 14.6i)T - 169iT^{2} \)
17 \( 1 - 28.0iT - 289T^{2} \)
19 \( 1 + (12.5 - 12.5i)T - 361iT^{2} \)
23 \( 1 + 29.2T + 529T^{2} \)
29 \( 1 + (-19.3 - 19.3i)T + 841iT^{2} \)
31 \( 1 - 11.6T + 961T^{2} \)
37 \( 1 + (0.771 + 0.771i)T + 1.36e3iT^{2} \)
41 \( 1 + 25.6T + 1.68e3T^{2} \)
43 \( 1 + (-40.5 - 40.5i)T + 1.84e3iT^{2} \)
47 \( 1 - 50.2iT - 2.20e3T^{2} \)
53 \( 1 + (46.2 - 46.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (22.7 - 22.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (12.7 - 12.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 122.T + 5.04e3T^{2} \)
73 \( 1 + 15.0iT - 5.32e3T^{2} \)
79 \( 1 + 51.3T + 6.24e3T^{2} \)
83 \( 1 + (-37.8 - 37.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 5.45T + 7.92e3T^{2} \)
97 \( 1 + 81.1T + 9.40e3T^{2} \)
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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

−10.99483894743439158697156649698, −10.69658137284367594448953276816, −9.918653254609980315401790799823, −8.438059484349651885727044716170, −7.62613282552275369725192301404, −6.33226958920535197483454794260, −5.79650204111706625289591386298, −4.33155677022440545507988724840, −3.51354836201917330484158650388, −1.30416726334722289148493871782, 0.47632914028539930661677401497, 2.18491369627934028763623282697, 4.09293244439004174608171956476, 5.00579883504016950251990549912, 6.15459247523119514822790461729, 6.86304114642849542895122249107, 8.105547184196621459990411297502, 9.024511851665351962225192993452, 10.07329621030304837337824023537, 11.15913037468163289066639269384

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