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Properties

Label 2-384-12.11-c3-0-26
Degree $2$
Conductor $384$
Sign $0.962 - 0.270i$
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

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(2.917573841\)
\(L(\frac12)\) \(\approx\) \(2.917573841\)
\(L(\frac{5}{2})\) 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

−10.91420620054117006750841734679, −9.876720517552197331601333312723, −9.040076099143572948643791055548, −8.287691658609985956139724670796, −7.11263558973446478693994026865, −6.53305149133922620135003191267, −4.98119977436445209060707672966, −3.60946414530650369021390500916, −2.71642753386457493272386781595, −1.31904055687935361821071118841, 1.08665625004247590926902300097, 2.54039315346895087860450569516, 3.90462277862726439729226917254, 4.63808370053142978604685906619, 6.12070904159274423949198908472, 7.32722871946558664913093827160, 8.192605909302343504006329081018, 9.101890018952369199548577866493, 9.701347489909407749239494562382, 10.78052770261745140243772210706

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