↑

Properties

Label 2-1440-5.2-c0-0-2
Degree $2$
Conductor $1440$
Sign $0.973 + 0.229i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
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  + 5-s + (1 − i)13-s + (−1 − i)17-s + 25-s + 2i·29-s + (1 + i)37-s − 2·41-s − i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 + i)73-s + (−1 − i)85-s + 2i·89-s + (−1 − i)97-s + 2i·109-s + ⋯
L(s)  = 1  + 5-s + (1 − i)13-s + (−1 − i)17-s + 25-s + 2i·29-s + (1 + i)37-s − 2·41-s − i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 + i)73-s + (−1 − i)85-s + 2i·89-s + (−1 − i)97-s + 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.315731624\)
\(L(\frac12)\) \(\approx\) \(1.315731624\)
\(L(1)\) 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 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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

−9.750468171704631929740761426681, −8.819825118749431193299193415884, −8.356620180901545215757275275054, −7.04522918523971434102280963918, −6.50447129242142655992783928096, −5.47128246144532710457912472558, −4.91304121345842909654633126375, −3.51573612738716185265442853736, −2.59686168500782016158249376578, −1.32639248226309068763830315054, 1.57524917341929851110694534915, 2.45924310097564594588447621042, 3.87085004248764801643048610709, 4.63430813009946904863747690975, 5.99698087262425325432497810660, 6.19668732674116076915441533266, 7.22003887251406042531982736089, 8.379783667186916001121314056584, 8.958377295751747429461176436404, 9.704891110795053292508302904530

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