Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.965 - 0.258i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4.38·5-s + i·7-s + 5.11i·11-s − 5.21i·13-s + 2i·17-s + 6.50·19-s + 4.63·23-s + 14.2·25-s + 1.26·29-s − 4.21i·31-s + 4.38i·35-s − 1.98i·37-s − 7.37i·41-s − 3.71·43-s − 2.57·47-s + ⋯
L(s)  = 1  + 1.96·5-s + 0.377i·7-s + 1.54i·11-s − 1.44i·13-s + 0.485i·17-s + 1.49·19-s + 0.967·23-s + 2.84·25-s + 0.235·29-s − 0.756i·31-s + 0.741i·35-s − 0.326i·37-s − 1.15i·41-s − 0.566·43-s − 0.375·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.965 - 0.258i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.965 - 0.258i)$
$L(1)$  $\approx$  $3.383481849$
$L(\frac12)$  $\approx$  $3.383481849$
$L(\frac{3}{2})$   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 \( 1 - 4.38T + 5T^{2} \)
11 \( 1 - 5.11iT - 11T^{2} \)
13 \( 1 + 5.21iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 6.50T + 19T^{2} \)
23 \( 1 - 4.63T + 23T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 + 4.21iT - 31T^{2} \)
37 \( 1 + 1.98iT - 37T^{2} \)
41 \( 1 + 7.37iT - 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 + 2.57T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 3.50iT - 59T^{2} \)
61 \( 1 + 4.53iT - 61T^{2} \)
67 \( 1 + 5.21T + 67T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 + 3.98iT - 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 - 4.42T + 97T^{2} \)
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

−8.072465236859452047361771064275, −7.32958186308090921697791044990, −6.58758616077691832326637896308, −5.90537441518223936409468573739, −5.13895521074228364535627318514, −4.98389784020655718023100551903, −3.46202422481423322843058727736, −2.61910101344009414843902186909, −1.95921825406442540299731609109, −1.07084058956806449728231798869, 1.03921693088365904270909527807, 1.64063992526293053522789000613, 2.83316748512958257491657654022, 3.29190441893512916146232250979, 4.70105123484283757082564850647, 5.20445894896794421484320520582, 5.98144210143069802449799107554, 6.54912281716281854738371855206, 7.07457854321502980076018781784, 8.154544375056189311259929437458

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