Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.944 - 0.327i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·5-s + (0.5 − 2.59i)7-s − 5.19i·11-s + 3.46i·13-s + 3.46i·17-s − 8·19-s + 6.92i·23-s + 2.00·25-s + 6·29-s + 5·31-s + (4.5 + 0.866i)35-s + 4·37-s + 6.92i·41-s − 3.46i·43-s + 6·47-s + ⋯
L(s)  = 1  + 0.774i·5-s + (0.188 − 0.981i)7-s − 1.56i·11-s + 0.960i·13-s + 0.840i·17-s − 1.83·19-s + 1.44i·23-s + 0.400·25-s + 1.11·29-s + 0.898·31-s + (0.760 + 0.146i)35-s + 0.657·37-s + 1.08i·41-s − 0.528i·43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.944 - 0.327i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.944 - 0.327i)\)
\(L(1)\)  \(\approx\)  \(1.772423858\)
\(L(\frac12)\)  \(\approx\)  \(1.772423858\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.504142080063773725374715809751, −8.191060283550992492951725543615, −7.13181084832077600128675820844, −6.46948651493064809398827198121, −6.02015333512657292130556185901, −4.74066408934052204389732941642, −3.93725464834244646603198141873, −3.27849515130741377685159542255, −2.14621413959342333315168179823, −0.893600883325719009879333510516, 0.74172155904591400381856507566, 2.20761604454694210534814596204, 2.68860226482550610841304715849, 4.35741577887307563580969055375, 4.67910710826956394426360723021, 5.51879362875736544522655995494, 6.41970057766840032573831491298, 7.14797674551729065439939757771, 8.216205831991562722195419660954, 8.587022818580158515072550805783

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