↑

Properties

Label 2-728-728.555-c0-0-0
Degree $2$
Conductor $728$
Sign $0.203 - 0.979i$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
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 + 3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s − 0.999·8-s − 0.999i·10-s + 11-s + (−0.499 + 0.866i)12-s + i·13-s + 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.499i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s − 0.999·8-s − 0.999i·10-s + 11-s + (−0.499 + 0.866i)12-s + i·13-s + 0.999i·14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 19-s + (0.866 − 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.203 - 0.979i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.435217657\)
\(L(\frac12)\) \(\approx\) \(1.435217657\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.08830000986947996829912201174, −9.246718335252239253062508643782, −8.770804179891043210404216125220, −8.368409693163070938685903664242, −7.39227713002733764076852229311, −6.55890819916875520630182972208, −5.26421446667596882623117464879, −4.31870346478650307923802958900, −3.66067459010521202077898548339, −2.20973621825204242855015424542, 1.56906326891980353279485018926, 3.01903489226972674555976396679, 3.63515027289522843831417384640, 4.54839496207433494790579845691, 5.74726058659740618990277102128, 7.14145515157524413221037468650, 7.924592507317041667693927539328, 8.830446296496800449208902868975, 9.516041875471359850239043153152, 10.80903322683203537553849327876

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