Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.972 - 0.232i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.512i·5-s + 7-s − 1.82i·11-s − 1.80i·13-s − 8.11·17-s + 3.43i·19-s − 3.65·23-s + 4.73·25-s + 7.98i·29-s + 1.56·31-s − 0.512i·35-s + 8.44i·37-s + 2.30·41-s − 10.7i·43-s + 11.3·47-s + ⋯
L(s)  = 1  − 0.229i·5-s + 0.377·7-s − 0.551i·11-s − 0.500i·13-s − 1.96·17-s + 0.789i·19-s − 0.762·23-s + 0.947·25-s + 1.48i·29-s + 0.281·31-s − 0.0866i·35-s + 1.38i·37-s + 0.360·41-s − 1.64i·43-s + 1.66·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.972 - 0.232i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.972 - 0.232i)$
$L(1)$  $\approx$  $1.752308248$
$L(\frac12)$  $\approx$  $1.752308248$
$L(\frac{3}{2})$   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 \( 1 - T \)
good5 \( 1 + 0.512iT - 5T^{2} \)
11 \( 1 + 1.82iT - 11T^{2} \)
13 \( 1 + 1.80iT - 13T^{2} \)
17 \( 1 + 8.11T + 17T^{2} \)
19 \( 1 - 3.43iT - 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 - 2.30T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 8.87iT - 53T^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 - 5.31iT - 61T^{2} \)
67 \( 1 - 6.44iT - 67T^{2} \)
71 \( 1 - 9.10T + 71T^{2} \)
73 \( 1 - 9.24T + 73T^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 + 4.53iT - 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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

−8.252728836447680882084134205085, −7.38253641697328047412565182296, −6.69022259296538925358983568528, −5.96408113802910659006064404454, −5.20618629316684384574847936101, −4.49827789529072480158014790934, −3.73477996929707872313764573829, −2.77408010821410357107079016993, −1.88683310098123621468421065098, −0.78295354212760430029894827203, 0.59920809127733180469888515757, 2.14203791786127777985349420726, 2.39471647809894149454908643624, 3.77270729705841322004835258428, 4.48418161969256402727171658533, 4.94800293035145080642788265130, 6.14349433718569690636006437254, 6.57736467235413818588822916796, 7.33557163459239884368224884538, 7.978126829766502062265186330321

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