Properties

Degree $2$
Conductor $3024$
Sign $-0.990 + 0.135i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.63i·5-s + (1 − 2.44i)7-s − 1.50i·11-s − 5.91i·13-s + 5.14i·17-s − 5.24·19-s − 8.78i·23-s − 8.24·25-s + 8.91·29-s − 3.24·31-s + (−8.91 − 3.63i)35-s + 37-s − 6.65i·41-s + 9.37i·43-s − 3.69·47-s + ⋯
L(s)  = 1  − 1.62i·5-s + (0.377 − 0.925i)7-s − 0.454i·11-s − 1.64i·13-s + 1.24i·17-s − 1.20·19-s − 1.83i·23-s − 1.64·25-s + 1.65·29-s − 0.582·31-s + (−1.50 − 0.615i)35-s + 0.164·37-s − 1.03i·41-s + 1.43i·43-s − 0.538·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.990 + 0.135i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513396310\)
\(L(\frac12)\) \(\approx\) \(1.513396310\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
good5 \( 1 + 3.63iT - 5T^{2} \)
11 \( 1 + 1.50iT - 11T^{2} \)
13 \( 1 + 5.91iT - 13T^{2} \)
17 \( 1 - 5.14iT - 17T^{2} \)
19 \( 1 + 5.24T + 19T^{2} \)
23 \( 1 + 8.78iT - 23T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 6.65iT - 41T^{2} \)
43 \( 1 - 9.37iT - 43T^{2} \)
47 \( 1 + 3.69T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + 0.420iT - 73T^{2} \)
79 \( 1 - 4.47iT - 79T^{2} \)
83 \( 1 - 3.69T + 83T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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

−8.412025291429080600902299996596, −8.018144151929269794329277376538, −6.82856033874754099043127073003, −5.95661307145539828861490768358, −5.21170423519660237421085735609, −4.40807325693662215342607286192, −3.89460281954542998628946986677, −2.53056085332672203852734687000, −1.19749245138678882116307096611, −0.50357090938047637364049492536, 1.87774330177259917140852766999, 2.46271910114166375219419299602, 3.40115118862095487011183889496, 4.38742206716570776523408786395, 5.28144398480595824472595878238, 6.26625124856155909717617274561, 6.84727111942860485937026068181, 7.35265042036705368821089106352, 8.330250316361748491776675050532, 9.202306100618040090175276249920

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