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  − 0.317i·5-s i·7-s + 4.87·11-s − 2.44·13-s − 0.317i·17-s + 0.449i·19-s − 2.82·23-s + 4.89·25-s + 7.70i·29-s + 4.44i·31-s − 0.317·35-s − 7·37-s + 5.97i·41-s + 10.3i·43-s − 3.92·47-s + ⋯
L(s)  = 1  − 0.142i·5-s − 0.377i·7-s + 1.47·11-s − 0.679·13-s − 0.0770i·17-s + 0.103i·19-s − 0.589·23-s + 0.979·25-s + 1.43i·29-s + 0.799i·31-s − 0.0537·35-s − 1.15·37-s + 0.933i·41-s + 1.57i·43-s − 0.572·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.857649949$
$L(\frac12)$  $\approx$  $1.857649949$
$L(\frac{3}{2})$   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 \( 1 + 0.317iT - 5T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 0.317iT - 17T^{2} \)
19 \( 1 - 0.449iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 7.70iT - 29T^{2} \)
31 \( 1 - 4.44iT - 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 - 5.97iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 3.92T + 47T^{2} \)
53 \( 1 + 0.635iT - 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 9.34T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 3.10T + 73T^{2} \)
79 \( 1 - 8.55iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.326877613460760039943885327644, −7.23786123676785579543620591051, −6.85108494761317547920514455321, −6.18038141401677628816405673133, −5.13802773802446783326470993454, −4.60634480901555214008836653305, −3.70640182948598609298679444543, −3.04338710699255371533246046680, −1.79573951910634519634507928737, −1.00396151521587073661790763150, 0.54378475954140263886431729626, 1.83312004451942824902122255991, 2.55513351075288154513406793566, 3.69798953415648458570642016016, 4.18222648238256787625575381062, 5.19559152802112171233755828042, 5.84378244811889077046147609868, 6.72071528993584967940918879640, 7.07506178211160351261941287207, 8.059838755896808069250910301273

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