Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.917 - 0.396i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.530·5-s i·7-s + 0.905i·11-s + 4.38i·13-s + 1.25i·17-s − 0.00364·19-s − 0.778·23-s − 4.71·25-s + 5.43·29-s − 3.49i·31-s + 0.530i·35-s + 5.34i·37-s − 1.74i·41-s + 0.259·43-s + 3.75·47-s + ⋯
L(s)  = 1  − 0.237·5-s − 0.377i·7-s + 0.272i·11-s + 1.21i·13-s + 0.304i·17-s − 0.000837·19-s − 0.162·23-s − 0.943·25-s + 1.00·29-s − 0.628i·31-s + 0.0895i·35-s + 0.878i·37-s − 0.272i·41-s + 0.0395·43-s + 0.548·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.917 - 0.396i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.917 - 0.396i)$
$L(1)$  $\approx$  $0.5078945075$
$L(\frac12)$  $\approx$  $0.5078945075$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 0.530T + 5T^{2} \)
11 \( 1 - 0.905iT - 11T^{2} \)
13 \( 1 - 4.38iT - 13T^{2} \)
17 \( 1 - 1.25iT - 17T^{2} \)
19 \( 1 + 0.00364T + 19T^{2} \)
23 \( 1 + 0.778T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 + 3.49iT - 31T^{2} \)
37 \( 1 - 5.34iT - 37T^{2} \)
41 \( 1 + 1.74iT - 41T^{2} \)
43 \( 1 - 0.259T + 43T^{2} \)
47 \( 1 - 3.75T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 5.73iT - 59T^{2} \)
61 \( 1 + 0.555iT - 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 1.96iT - 79T^{2} \)
83 \( 1 - 5.82iT - 83T^{2} \)
89 \( 1 - 1.94iT - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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

−8.359501133922020662618300794079, −7.62657007105792623940890218194, −6.99964218793579890938787117647, −6.32895152578841529058316440838, −5.60214854996155587308609467201, −4.46337132277831322069424058748, −4.22991890641166154426263118480, −3.21564224636961907834829065413, −2.19777440012886365387940156176, −1.30567106949656293449875793487, 0.13230541179132730607373874619, 1.32146007378122249643601002097, 2.56031141544875883552891265246, 3.16788032200858322029506373491, 4.09465335906014947334258718164, 4.90809625334516171921343225051, 5.74447490777372780993664565599, 6.14069831943445985433205796346, 7.28510511715512386113692443670, 7.66180824007601582782940512820

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