Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.377 + 0.925i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·5-s + (−1 − 2.44i)7-s − 1.41i·11-s + 2.44i·13-s + 5.19·17-s + 7.34i·19-s + 2.82i·23-s − 2.00·25-s − 7.07i·29-s − 2.44i·31-s + (1.73 + 4.24i)35-s + 5·37-s + 8.66·41-s − 5·43-s + 8.66·47-s + ⋯
L(s)  = 1  − 0.774·5-s + (−0.377 − 0.925i)7-s − 0.426i·11-s + 0.679i·13-s + 1.26·17-s + 1.68i·19-s + 0.589i·23-s − 0.400·25-s − 1.31i·29-s − 0.439i·31-s + (0.292 + 0.717i)35-s + 0.821·37-s + 1.35·41-s − 0.762·43-s + 1.26·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.377 + 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.377 + 0.925i)\)
\(L(1)\)  \(\approx\)  \(1.241360044\)
\(L(\frac12)\)  \(\approx\)  \(1.241360044\)
\(L(\frac{3}{2})\)   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 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 + 2.44iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 8.66T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 2.44iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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

−8.305467701370379544618459082287, −7.78760632315514399560430889462, −7.33173853232825300273092888702, −6.21397935896695002107825542070, −5.71489816820015842887008912033, −4.39397653356963753801259640463, −3.85211206974430823711303568972, −3.19701259060450541425572749375, −1.71976668654173784339913194175, −0.50176122247856253311475649149, 0.946872340514202530897716208534, 2.53391645658292052249275060187, 3.11923114137449066387456638536, 4.16310535481772953083875700005, 5.06758980826642576927217905432, 5.73531807293022851674778658168, 6.66765223256726731678443406938, 7.46239140112011515492787183628, 8.040633312890816906626301760062, 8.952491534193069840731753478085

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