↑

Properties

Label 2-2888-152.83-c0-0-6
Degree $2$
Conductor $2888$
Sign $-0.0977 + 0.995i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
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)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s − 27-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)33-s + (0.999 − 1.73i)34-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 + 0.866i)25-s − 27-s + (0.499 − 0.866i)32-s + (0.5 + 0.866i)33-s + (0.999 − 1.73i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5650903865\)
\(L(\frac12)\) \(\approx\) \(0.5650903865\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.580028091241895276800817583341, −7.60600050595495350957093875903, −7.14063644780738912189150773393, −6.66308065809917527518560737244, −5.60294529812172190345556724898, −5.22681297632619147191299506100, −4.22222349083784377532298383670, −3.15503398177150434446826026739, −2.12332518763700143471709127563, −0.30056513078845589079011173225, 1.70433630472268427895816929752, 2.66637367385186739418632397740, 3.76693546526579314090881701803, 4.44488182282661863248818110047, 5.02092706577844635355845019888, 5.89848079136526335468119381254, 6.47111389318349038948290004592, 7.87831424170390515962659783632, 8.518084472217925575818419783092, 9.501136080335641113764785390223

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