↑

Properties

Label 2-3024-7.5-c0-0-0
Degree $2$
Conductor $3024$
Sign $0.795 + 0.605i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + 7-s − 1.73i·13-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 43-s + 49-s + (1.5 + 0.866i)61-s + (−0.5 − 0.866i)67-s + (−0.5 + 0.866i)79-s − 1.73i·91-s + 1.73i·97-s + (1.5 + 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  + 7-s − 1.73i·13-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 43-s + 49-s + (1.5 + 0.866i)61-s + (−0.5 − 0.866i)67-s + (−0.5 + 0.866i)79-s − 1.73i·91-s + 1.73i·97-s + (1.5 + 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.795 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.374789308\)
\(L(\frac12)\) \(\approx\) \(1.374789308\)
\(L(1)\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.461873063680084174772506232162, −8.210396102352423138216133172160, −7.52804843762181266144291761352, −6.52180761045433012056834322882, −5.68610442572027824444139252728, −5.01267056771897243770696118081, −4.21365371673157062921686257434, −3.14130252537827244101300688766, −2.23824506610150294948879339969, −0.947690568544799367276752240675, 1.44705810549119933404593960722, 2.21163386245181703584064740596, 3.50939181976400416816343747200, 4.41592211344093801552206808659, 4.99154630863048266366985266537, 5.94795617781638364497098502006, 6.85500296758258296281903152674, 7.39520608821714860115874274368, 8.390496838673656267932891779892, 8.827445410240622499585937216638

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