Properties

Degree $2$
Conductor $384$
Sign $-i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 2.82·5-s + 4.89i·7-s − 2.99·9-s + 3.46i·11-s + 4.89i·15-s + 8.48·21-s + 3.00·25-s + 5.19i·27-s − 2.82·29-s + 4.89i·31-s + 5.99·33-s − 13.8i·35-s + 8.48·45-s − 16.9·49-s + ⋯
L(s)  = 1  − 0.999i·3-s − 1.26·5-s + 1.85i·7-s − 0.999·9-s + 1.04i·11-s + 1.26i·15-s + 1.85·21-s + 0.600·25-s + 0.999i·27-s − 0.525·29-s + 0.879i·31-s + 1.04·33-s − 2.34i·35-s + 1.26·45-s − 2.42·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-i$
Motivic weight: \(1\)
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459238 + 0.459238i\)
\(L(\frac12)\) \(\approx\) \(0.459238 + 0.459238i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 - 4.89iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−11.85391895752036835656619034023, −11.06068753562668365735021918709, −9.469298614558744411368411244442, −8.596078662628235903694244436086, −7.87483765279035490706106319678, −6.97879086546311414503732247931, −5.89490986982345546595031576565, −4.80642891446490399157318453175, −3.17875123777012981963514068096, −1.98282690862384287691381763601, 0.42577777916060539950582537275, 3.41097797971025020596928136546, 3.91966465286361320681899352479, 4.86626467305673063565261434773, 6.36251806673901999294985814587, 7.62000102677691629712604027683, 8.171050093744916000144528180532, 9.394604417873036672274040127806, 10.38799510489653323889677809741, 11.12138546634818983593310229440

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