Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.01i·5-s − 2.64·7-s − 2.01i·11-s − 7.34i·17-s − 3.35·19-s + 9.36i·23-s + 0.937·25-s + 2.29·31-s − 5.33i·35-s + 11.9·37-s − 9.36i·41-s + 7.00·49-s + 4.06·55-s − 16.7i·71-s + 5.33i·77-s + ⋯
L(s)  = 1  + 0.901i·5-s − 0.999·7-s − 0.607i·11-s − 1.78i·17-s − 0.769·19-s + 1.95i·23-s + 0.187·25-s + 0.411·31-s − 0.901i·35-s + 1.96·37-s − 1.46i·41-s + 49-s + 0.547·55-s − 1.98i·71-s + 0.607i·77-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.423413890\)
\(L(\frac12)\) \(\approx\) \(1.423413890\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good5 \( 1 - 2.01iT - 5T^{2} \)
11 \( 1 + 2.01iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 + 3.35T + 19T^{2} \)
23 \( 1 - 9.36iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + 9.36iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 16.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−8.942361838760175885343664376163, −7.73290384044356327482037142158, −7.22953840896219303200986513283, −6.46054152224062639764582792566, −5.85580522876473977515431457918, −4.91200924686703184812761832225, −3.73244079010820176763374269038, −3.09350204979369570807572078654, −2.35202285888336777694489587132, −0.64939869264194115234745103004, 0.78213556610527361847047356801, 2.05333297566573759586525239512, 3.05475326010469232405578264598, 4.36131327197173892145004326711, 4.47479126498541948284134646127, 5.91693641559906708360944033214, 6.29519061502422504097346805851, 7.13942039037987873125192010254, 8.383866320236161089818891693052, 8.454987433426359864865703047937

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