Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0574 - 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·5-s + i·7-s − 2.84i·11-s − 0.223i·13-s + 0.397i·17-s + 4.64·19-s − 7.36·23-s − 3.36·25-s + 10.6·29-s + 7.67i·31-s − 1.27i·35-s + 4.74i·37-s − 1.97i·41-s − 7.62·43-s − 11.0·47-s + ⋯
L(s)  = 1  − 0.571·5-s + 0.377i·7-s − 0.858i·11-s − 0.0620i·13-s + 0.0964i·17-s + 1.06·19-s − 1.53·23-s − 0.673·25-s + 1.97·29-s + 1.37i·31-s − 0.216i·35-s + 0.780i·37-s − 0.308i·41-s − 1.16·43-s − 1.61·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0574 - 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0574 - 0.998i)$
$L(1)$  $\approx$  $1.040279889$
$L(\frac12)$  $\approx$  $1.040279889$
$L(\frac{3}{2})$   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 \( 1 + 1.27T + 5T^{2} \)
11 \( 1 + 2.84iT - 11T^{2} \)
13 \( 1 + 0.223iT - 13T^{2} \)
17 \( 1 - 0.397iT - 17T^{2} \)
19 \( 1 - 4.64T + 19T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 7.67iT - 31T^{2} \)
37 \( 1 - 4.74iT - 37T^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 + 7.62T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 0.295T + 53T^{2} \)
59 \( 1 + 7.25iT - 59T^{2} \)
61 \( 1 + 9.45iT - 61T^{2} \)
67 \( 1 - 3.52T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 5.22iT - 79T^{2} \)
83 \( 1 - 9.11iT - 83T^{2} \)
89 \( 1 + 8.94iT - 89T^{2} \)
97 \( 1 - 0.228T + 97T^{2} \)
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

−8.192180093866048794693635023591, −7.82065843262633332871750848497, −6.65003585125937161507119861622, −6.31016968824564963267404319971, −5.27295757628931176944681574818, −4.78411436434469693522550744598, −3.59922769702337509688170733802, −3.25460582895426218266128027842, −2.11437499139526113947796320718, −0.966833375167859802316198198063, 0.31058429216505127774617688533, 1.57262730493874515672700437976, 2.54965714580771312410270745317, 3.55416018536895139610445227224, 4.23599791452400809828081531045, 4.85308698504200538002327762157, 5.79093886841293946581859405245, 6.55978099026365103984114232700, 7.26955496860845326176030030681, 7.926791655194202771884609152693

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