↑

Properties

Label 2-3888-9.5-c0-0-0
Degree $2$
Conductor $3888$
Sign $-0.939 - 0.342i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)7-s + (−1 + 1.73i)13-s − 2·19-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s − 37-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)61-s + (−0.5 + 0.866i)67-s − 73-s + (1 + 1.73i)79-s + 1.99·91-s + (−1 − 1.73i)97-s + (−0.5 + 0.866i)103-s − 109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−1 + 1.73i)13-s − 2·19-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s − 37-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)61-s + (−0.5 + 0.866i)67-s − 73-s + (1 + 1.73i)79-s + 1.99·91-s + (−1 − 1.73i)97-s + (−0.5 + 0.866i)103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3888} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1347155832\)
\(L(\frac12)\) \(\approx\) \(0.1347155832\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.886808015529885984808044328872, −8.402467183063853393728792728529, −7.22939508447414411011579154741, −6.88121592284180718574654588969, −6.26321941633441879256100865509, −5.12933924509255600012966557161, −4.23420722478004079727463646490, −3.88429480254012451373613415065, −2.53149522541700497962983290257, −1.73211871609483328761125436127, 0.06820835859080096726495461965, 1.94309415356894074357782550771, 2.74842833700036690538867205464, 3.54376519921051401387797530369, 4.62913795408109624020504261532, 5.45078450852989825533104866316, 6.00489340793297580582579977728, 6.80933246713742258843350578098, 7.74440841320050397953276222343, 8.252037828279487126816364464705

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