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Properties

Label 2-384-12.11-c3-0-21
Degree $2$
Conductor $384$
Sign $0.270 - 0.962i$
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  + (5.00 + 1.40i)3-s + 5.86i·5-s + 5.92i·7-s + (23.0 + 14.0i)9-s + 27.9·11-s + 0.0653·13-s + (−8.26 + 29.3i)15-s + 36.9i·17-s − 30.7i·19-s + (−8.33 + 29.6i)21-s − 61.2·23-s + 90.5·25-s + (95.3 + 102. i)27-s + 143. i·29-s + 299. i·31-s + ⋯
L(s)  = 1  + (0.962 + 0.270i)3-s + 0.524i·5-s + 0.319i·7-s + (0.853 + 0.521i)9-s + 0.766·11-s + 0.00139·13-s + (−0.142 + 0.505i)15-s + 0.527i·17-s − 0.371i·19-s + (−0.0866 + 0.307i)21-s − 0.555·23-s + 0.724·25-s + (0.679 + 0.733i)27-s + 0.919i·29-s + 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.270 - 0.962i$
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: \((2,\ 384,\ (\ :3/2),\ 0.270 - 0.962i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.734098447\)
\(L(\frac12)\) \(\approx\) \(2.734098447\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.00 - 1.40i)T \)
good5 \( 1 - 5.86iT - 125T^{2} \)
7 \( 1 - 5.92iT - 343T^{2} \)
11 \( 1 - 27.9T + 1.33e3T^{2} \)
13 \( 1 - 0.0653T + 2.19e3T^{2} \)
17 \( 1 - 36.9iT - 4.91e3T^{2} \)
19 \( 1 + 30.7iT - 6.85e3T^{2} \)
23 \( 1 + 61.2T + 1.21e4T^{2} \)
29 \( 1 - 143. iT - 2.43e4T^{2} \)
31 \( 1 - 299. iT - 2.97e4T^{2} \)
37 \( 1 + 340.T + 5.06e4T^{2} \)
41 \( 1 + 379. iT - 6.89e4T^{2} \)
43 \( 1 - 470. iT - 7.95e4T^{2} \)
47 \( 1 - 428.T + 1.03e5T^{2} \)
53 \( 1 + 505. iT - 1.48e5T^{2} \)
59 \( 1 + 207.T + 2.05e5T^{2} \)
61 \( 1 + 578.T + 2.26e5T^{2} \)
67 \( 1 - 415. iT - 3.00e5T^{2} \)
71 \( 1 - 547.T + 3.57e5T^{2} \)
73 \( 1 - 194.T + 3.89e5T^{2} \)
79 \( 1 - 308. iT - 4.93e5T^{2} \)
83 \( 1 - 62.3T + 5.71e5T^{2} \)
89 \( 1 + 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 - 703.T + 9.12e5T^{2} \)
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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

−10.83339001080501386304533284929, −10.23499057282611727375272283168, −9.041481835242417330784293680925, −8.609156326172113413529147393174, −7.33944687741080973685027689450, −6.57199339661145951888962321448, −5.12188497306604653252042404850, −3.86888878760960006717777879688, −2.94037133926162978876221431379, −1.62947085218357318018989685365, 0.881198934719002386603170479656, 2.21056116957187655844178864332, 3.63518850746307485447702628547, 4.52412618624002249705923150027, 6.02839813828683898921741922335, 7.14557332050111540293921164819, 7.970503072857723446697938598312, 8.932210349222188768943202173193, 9.557736806423588049801633989633, 10.57747161599814025127006934074

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