↑

Properties

Label 2-880-55.54-c0-0-2
Degree $2$
Conductor $880$
Sign $-0.5 + 0.866i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  − 1.73i·3-s + (0.5 − 0.866i)5-s − 1.99·9-s + 11-s + (−1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 − 0.866i)25-s + 1.73i·27-s − 31-s − 1.73i·33-s + 1.73i·37-s + (−0.999 + 1.73i)45-s − 49-s + (0.5 − 0.866i)55-s + 59-s + ⋯
L(s)  = 1  − 1.73i·3-s + (0.5 − 0.866i)5-s − 1.99·9-s + 11-s + (−1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 − 0.866i)25-s + 1.73i·27-s − 31-s − 1.73i·33-s + 1.73i·37-s + (−0.999 + 1.73i)45-s − 49-s + (0.5 − 0.866i)55-s + 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069081048\)
\(L(\frac12)\) \(\approx\) \(1.069081048\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.816064417801211431424999679181, −9.075804793792927686631635357774, −8.275323252312199843062904256573, −7.50702262269627597452826470860, −6.61612082254370158783954504007, −5.93806298026957659300460429578, −4.99592629166217049725002194104, −3.47037170613842761347265608124, −1.96335331163337255569281907212, −1.22620975675728170239341745463, 2.38218283205467525109772655125, 3.52942851288700712550693422582, 4.21205817982189336098735354918, 5.29324465123950895712765949971, 6.15893898753683633052192165913, 7.05460101415433733235848336056, 8.448033777754738876352687834600, 9.241654603129487465488687566121, 9.777571143693556337217260708940, 10.64885043816639158555759927902

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