↑

Properties

Label 2-384-12.11-c5-0-38
Degree $2$
Conductor $384$
Sign $-0.118 - 0.992i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (15.4 − 1.85i)3-s + 55.8i·5-s + 225. i·7-s + (236. − 57.4i)9-s − 36.9·11-s + 795.·13-s + (103. + 864. i)15-s − 129. i·17-s − 998. i·19-s + (419. + 3.49e3i)21-s + 3.74e3·23-s + 4.87·25-s + (3.54e3 − 1.32e3i)27-s + 921. i·29-s + 423. i·31-s + ⋯
L(s)  = 1  + (0.992 − 0.118i)3-s + 0.999i·5-s + 1.74i·7-s + (0.971 − 0.236i)9-s − 0.0920·11-s + 1.30·13-s + (0.118 + 0.992i)15-s − 0.108i·17-s − 0.634i·19-s + (0.207 + 1.73i)21-s + 1.47·23-s + 0.00156·25-s + (0.936 − 0.350i)27-s + 0.203i·29-s + 0.0791i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.364946364\)
\(L(\frac12)\) \(\approx\) \(3.364946364\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.4 + 1.85i)T \)
good5 \( 1 - 55.8iT - 3.12e3T^{2} \)
7 \( 1 - 225. iT - 1.68e4T^{2} \)
11 \( 1 + 36.9T + 1.61e5T^{2} \)
13 \( 1 - 795.T + 3.71e5T^{2} \)
17 \( 1 + 129. iT - 1.41e6T^{2} \)
19 \( 1 + 998. iT - 2.47e6T^{2} \)
23 \( 1 - 3.74e3T + 6.43e6T^{2} \)
29 \( 1 - 921. iT - 2.05e7T^{2} \)
31 \( 1 - 423. iT - 2.86e7T^{2} \)
37 \( 1 + 1.09e4T + 6.93e7T^{2} \)
41 \( 1 - 1.44e4iT - 1.15e8T^{2} \)
43 \( 1 - 2.08e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.20e4T + 2.29e8T^{2} \)
53 \( 1 + 1.13e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.66e4T + 7.14e8T^{2} \)
61 \( 1 - 3.82e4T + 8.44e8T^{2} \)
67 \( 1 + 8.97e3iT - 1.35e9T^{2} \)
71 \( 1 + 6.16e4T + 1.80e9T^{2} \)
73 \( 1 - 2.56e3T + 2.07e9T^{2} \)
79 \( 1 - 6.15e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + 1.25e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.56e5T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.84056713654164396008518556803, −9.606346899270271906958661368341, −8.853263008295832412187003323569, −8.224770996089533362011575700235, −6.95196461082560525944524591286, −6.20643870175843519188818581836, −4.90387798329916513445651440427, −3.22418049740974667809664299679, −2.80063425711528216238986829262, −1.53729162258342359083102320702, 0.77182573564791675025047920324, 1.58039093437725584540906844800, 3.43914367731803382672686724140, 4.07677143679017047116898751702, 5.13127754819392542451327397868, 6.75038608946637541006636731100, 7.59462780687435127992012689843, 8.528551218302194639809042124653, 9.139639531833290289839808326427, 10.35263246014359964034676769183

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