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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.517i·5-s i·7-s − 3.18·11-s − 2.64·13-s − 5.15i·17-s + 8.23i·19-s + 4.38·23-s + 4.73·25-s + 5.11i·29-s − 0.548i·31-s − 0.517·35-s − 2.37·37-s − 0.473i·41-s + 0.816i·43-s + 0.493·47-s + ⋯
L(s)  = 1  − 0.231i·5-s − 0.377i·7-s − 0.959·11-s − 0.734·13-s − 1.25i·17-s + 1.88i·19-s + 0.913·23-s + 0.946·25-s + 0.949i·29-s − 0.0985i·31-s − 0.0874·35-s − 0.391·37-s − 0.0740i·41-s + 0.124i·43-s + 0.0719·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $1.539761789$
$L(\frac12)$  $\approx$  $1.539761789$
$L(\frac{3}{2})$   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 \( 1 + 0.517iT - 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + 5.15iT - 17T^{2} \)
19 \( 1 - 8.23iT - 19T^{2} \)
23 \( 1 - 4.38T + 23T^{2} \)
29 \( 1 - 5.11iT - 29T^{2} \)
31 \( 1 + 0.548iT - 31T^{2} \)
37 \( 1 + 2.37T + 37T^{2} \)
41 \( 1 + 0.473iT - 41T^{2} \)
43 \( 1 - 0.816iT - 43T^{2} \)
47 \( 1 - 0.493T + 47T^{2} \)
53 \( 1 - 0.915iT - 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 3.40T + 61T^{2} \)
67 \( 1 + 0.585iT - 67T^{2} \)
71 \( 1 - 6.95T + 71T^{2} \)
73 \( 1 - 2.41T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + 0.215T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + 7.23T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.900870028924827034338584918732, −7.30500710252296923099393394139, −6.75921824900619446476213478325, −5.65276316496305574298389346721, −5.12518308540516624917301679268, −4.49871292210457718994023970863, −3.41512734118491784343016848867, −2.76756182538001483210510790511, −1.69220839994654039657814650829, −0.53554948187952728603564943825, 0.77303807237320767883596877030, 2.26182831360569238063104261590, 2.69746400329521398233874608945, 3.67052902368469923799770817070, 4.77979936346422757824977636427, 5.13248380955747175802167775303, 6.05260284157973274056842048243, 6.86541821903390986199681212282, 7.33160126237009352741901101035, 8.273095105200836604786221659894

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