Properties

Label 2-384-24.11-c3-0-40
Degree $2$
Conductor $384$
Sign $-0.557 + 0.829i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.09 − i)3-s − 10.1·5-s − 10.1i·7-s + (24.9 − 10.1i)9-s − 46i·11-s + 44i·13-s + (−51.9 + 10.1i)15-s − 20.3i·17-s − 10.1·19-s + (−10.1 − 51.9i)21-s − 88·23-s − 21.0·25-s + (117. − 76.9i)27-s − 254.·29-s − 214. i·31-s + ⋯
L(s)  = 1  + (0.981 − 0.192i)3-s − 0.912·5-s − 0.550i·7-s + (0.925 − 0.377i)9-s − 1.26i·11-s + 0.938i·13-s + (−0.895 + 0.175i)15-s − 0.290i·17-s − 0.123·19-s + (−0.105 − 0.540i)21-s − 0.797·23-s − 0.168·25-s + (0.835 − 0.548i)27-s − 1.63·29-s − 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.557 + 0.829i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.557 + 0.829i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.527169044\)
\(L(\frac12)\) \(\approx\) \(1.527169044\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.09 + i)T \)
good5 \( 1 + 10.1T + 125T^{2} \)
7 \( 1 + 10.1iT - 343T^{2} \)
11 \( 1 + 46iT - 1.33e3T^{2} \)
13 \( 1 - 44iT - 2.19e3T^{2} \)
17 \( 1 + 20.3iT - 4.91e3T^{2} \)
19 \( 1 + 10.1T + 6.85e3T^{2} \)
23 \( 1 + 88T + 1.21e4T^{2} \)
29 \( 1 + 254.T + 2.43e4T^{2} \)
31 \( 1 + 214. iT - 2.97e4T^{2} \)
37 \( 1 + 332iT - 5.06e4T^{2} \)
41 \( 1 + 489. iT - 6.89e4T^{2} \)
43 \( 1 + 234.T + 7.95e4T^{2} \)
47 \( 1 - 384T + 1.03e5T^{2} \)
53 \( 1 + 458.T + 1.48e5T^{2} \)
59 \( 1 - 630iT - 2.05e5T^{2} \)
61 \( 1 - 236iT - 2.26e5T^{2} \)
67 \( 1 - 50.9T + 3.00e5T^{2} \)
71 \( 1 - 680T + 3.57e5T^{2} \)
73 \( 1 - 422T + 3.89e5T^{2} \)
79 \( 1 + 744. iT - 4.93e5T^{2} \)
83 \( 1 - 186iT - 5.71e5T^{2} \)
89 \( 1 + 958. iT - 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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

−10.68393481816222054687915572992, −9.434366048993746816577986261505, −8.724395204025057829375286997959, −7.73739291001308760953478211897, −7.19402619003018246735693884172, −5.85897032405063957481865082842, −4.07306887566258519281176068495, −3.69114182946198317984541354092, −2.13709432572039998663957880386, −0.44340141307865287955654488317, 1.80274762545815569547717434503, 3.10635570455659761780372494620, 4.11110490410458870102468278499, 5.16036968192028427278117327986, 6.73628133027435979123325019843, 7.84097416359218164220247891634, 8.217859389518921427544262008722, 9.435784136285047353029702017586, 10.10033000374051252352382487373, 11.21519487136621283514015385632

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