Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 + 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.29i·5-s + i·7-s + 2.21·11-s − 1.03·13-s + 2.24i·17-s − 8.63i·19-s − 8.96·23-s + 3.31·25-s − 3.26i·29-s − 4.37i·31-s − 1.29·35-s − 7.21·37-s + 7.58i·41-s + 12.9i·43-s + 5.24·47-s + ⋯
L(s)  = 1  + 0.579i·5-s + 0.377i·7-s + 0.668·11-s − 0.287·13-s + 0.544i·17-s − 1.98i·19-s − 1.87·23-s + 0.663·25-s − 0.606i·29-s − 0.786i·31-s − 0.219·35-s − 1.18·37-s + 1.18i·41-s + 1.97i·43-s + 0.764·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$
$L(1)$  $\approx$  $0.3702224982$
$L(\frac12)$  $\approx$  $0.3702224982$
$L(\frac{3}{2})$   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 \( 1 - 1.29iT - 5T^{2} \)
11 \( 1 - 2.21T + 11T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 - 2.24iT - 17T^{2} \)
19 \( 1 + 8.63iT - 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 + 3.26iT - 29T^{2} \)
31 \( 1 + 4.37iT - 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 - 7.58iT - 41T^{2} \)
43 \( 1 - 12.9iT - 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 + 2.46T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 4.46iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 + 9.14iT - 79T^{2} \)
83 \( 1 + 9.96T + 83T^{2} \)
89 \( 1 + 3.48iT - 89T^{2} \)
97 \( 1 + 6.30T + 97T^{2} \)
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

−7.75243726894612986445984640446, −7.07453963117820437290956159500, −6.32147823908891304759881344587, −5.91964699063883194998549111258, −4.76204398846779873231164725098, −4.25352024874740784815551661995, −3.18586076785750390816434074231, −2.53066727161536138261165622270, −1.56379560688370656229154166118, −0.091510857620809956142561656216, 1.27111877920716736440291527206, 1.98897830281905938938721592325, 3.30255104068881651260923813735, 3.96527524043900104827764165326, 4.64046811047949697294948604178, 5.60782214853045053437342401831, 6.03708364672826464619880250107, 7.15544794508325632210044718934, 7.46352516454879161088853755254, 8.578971027197203215894820556953

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