↑

Properties

Label 2-3024-7.6-c0-0-1
Degree $2$
Conductor $3024$
Sign $-i$
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  + 1.73i·5-s + 7-s + 1.73i·17-s − 1.99·25-s + 1.73i·35-s + 37-s − 1.73i·41-s − 43-s − 1.73i·47-s + 49-s + 1.73i·59-s − 2·67-s + 79-s + 1.73i·83-s − 2.99·85-s + ⋯
L(s)  = 1  + 1.73i·5-s + 7-s + 1.73i·17-s − 1.99·25-s + 1.73i·35-s + 37-s − 1.73i·41-s − 43-s − 1.73i·47-s + 49-s + 1.73i·59-s − 2·67-s + 79-s + 1.73i·83-s − 2.99·85-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337903873\)
\(L(\frac12)\) \(\approx\) \(1.337903873\)
\(L(1)\) 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 \( 1 - 1.73iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.73iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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

−8.971434214457733538101075978741, −8.225444421599884955409110281250, −7.54683341707472553299442476092, −6.88813098483145403202263506948, −6.12544589897098608098052640703, −5.43591481680323737437400003679, −4.20440043296252529937990751821, −3.57530321179120267223869669870, −2.51226482984222589948996111273, −1.71811150765318062888341207723, 0.874020922461783655717218912061, 1.78692070865236446985622004231, 3.03162559877840238268089610469, 4.47784437848657049244926002449, 4.71328834798529393613601967817, 5.40786659483238764739364568965, 6.33423965761509016226875554844, 7.56732748765223462361107899286, 7.956956045757634347067280164300, 8.749545310839394340852514260220

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