Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.994 - 0.108i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.162·5-s + i·7-s − 2.40i·11-s + 4.76i·13-s − 2.74i·17-s + 1.86·19-s + 1.32·23-s − 4.97·25-s + 3.96·29-s − 8.01i·31-s + 0.162i·35-s + 6.09i·37-s − 3.16i·41-s + 1.70·43-s + 12.1·47-s + ⋯
L(s)  = 1  + 0.0724·5-s + 0.377i·7-s − 0.724i·11-s + 1.32i·13-s − 0.665i·17-s + 0.426·19-s + 0.277·23-s − 0.994·25-s + 0.735·29-s − 1.43i·31-s + 0.0273i·35-s + 1.00i·37-s − 0.494i·41-s + 0.260·43-s + 1.77·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.994 - 0.108i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.994 - 0.108i)$
$L(1)$  $\approx$  $1.936308224$
$L(\frac12)$  $\approx$  $1.936308224$
$L(\frac{3}{2})$   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 \( 1 - 0.162T + 5T^{2} \)
11 \( 1 + 2.40iT - 11T^{2} \)
13 \( 1 - 4.76iT - 13T^{2} \)
17 \( 1 + 2.74iT - 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
23 \( 1 - 1.32T + 23T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 + 8.01iT - 31T^{2} \)
37 \( 1 - 6.09iT - 37T^{2} \)
41 \( 1 + 3.16iT - 41T^{2} \)
43 \( 1 - 1.70T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 8.22T + 53T^{2} \)
59 \( 1 + 5.45iT - 59T^{2} \)
61 \( 1 - 4.28iT - 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 6.37T + 73T^{2} \)
79 \( 1 + 9.75iT - 79T^{2} \)
83 \( 1 - 16.4iT - 83T^{2} \)
89 \( 1 - 2.35iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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

−8.083175005890027409236675747456, −7.40392138293381456617817528484, −6.59343940776711009177062674189, −6.01218828740992335167534807125, −5.25533104305342782130927692921, −4.43870262470863864044139722785, −3.68779229437927462036120586692, −2.72620380515324566840891011697, −1.95222047008420591932174812245, −0.73664487090098946585606027851, 0.73420279294321303951971219508, 1.79010178076512566671565412106, 2.82627235015425540350587982762, 3.60578433236467347657302025903, 4.43055025739323112984750281139, 5.23348056547866613640805161530, 5.87349473945451243622011503461, 6.69530130791490349966509592280, 7.46912172802507451303463318559, 7.923304634790753890165650994295

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