Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0820 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.99·5-s i·7-s − 5.36i·11-s + 4.55i·13-s + 0.958i·17-s − 0.532·19-s + 2.78·23-s + 3.98·25-s − 5.15·29-s + 4.66i·31-s + 2.99i·35-s − 6.10i·37-s − 12.3i·41-s − 4.45·43-s + 1.40·47-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377i·7-s − 1.61i·11-s + 1.26i·13-s + 0.232i·17-s − 0.122·19-s + 0.581·23-s + 0.796·25-s − 0.957·29-s + 0.837i·31-s + 0.506i·35-s − 1.00i·37-s − 1.92i·41-s − 0.679·43-s + 0.204·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0820 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0820 - 0.996i)$
$L(1)$  $\approx$  $0.5100782741$
$L(\frac12)$  $\approx$  $0.5100782741$
$L(\frac{3}{2})$   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 \( 1 + 2.99T + 5T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 - 4.55iT - 13T^{2} \)
17 \( 1 - 0.958iT - 17T^{2} \)
19 \( 1 + 0.532T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 + 5.15T + 29T^{2} \)
31 \( 1 - 4.66iT - 31T^{2} \)
37 \( 1 + 6.10iT - 37T^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + 4.45T + 43T^{2} \)
47 \( 1 - 1.40T + 47T^{2} \)
53 \( 1 + 9.57T + 53T^{2} \)
59 \( 1 + 0.0356iT - 59T^{2} \)
61 \( 1 + 5.13iT - 61T^{2} \)
67 \( 1 + 8.59T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 + 1.90iT - 79T^{2} \)
83 \( 1 + 3.13iT - 83T^{2} \)
89 \( 1 - 5.91iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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

−8.251348967197320247752606976128, −7.55856732620978947103442674773, −6.95173587225709701466019058560, −6.24735463339282905198832216383, −5.34604131592650494260093295220, −4.50847813665941966317267550141, −3.59986125122271389058536814154, −3.49320125451843986545112103084, −2.08066606567912451225111729454, −0.829137205884351128877993472752, 0.17345012531038593828825871023, 1.53793029119685332514583253212, 2.71166873379877481183959677701, 3.38760761296297749152305054613, 4.35048453081909068648248070972, 4.82363197748012797492920972734, 5.66214368879602515747603418355, 6.66276835419229007024957640452, 7.29635294998711200188334368492, 7.974072490260432784361105508677

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