Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.642 - 0.766i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.863·5-s i·7-s + 2.62i·11-s + 1.01i·13-s − 2.34i·17-s + 4.09·19-s + 6.23·23-s − 4.25·25-s + 1.57·29-s + 6.29i·31-s + 0.863i·35-s + 5.18i·37-s − 10.1i·41-s + 0.496·43-s − 5.24·47-s + ⋯
L(s)  = 1  − 0.386·5-s − 0.377i·7-s + 0.792i·11-s + 0.282i·13-s − 0.568i·17-s + 0.939·19-s + 1.29·23-s − 0.850·25-s + 0.292·29-s + 1.13i·31-s + 0.145i·35-s + 0.851i·37-s − 1.58i·41-s + 0.0756·43-s − 0.764·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.642 - 0.766i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.642 - 0.766i)$
$L(1)$  $\approx$  $1.605347596$
$L(\frac12)$  $\approx$  $1.605347596$
$L(\frac{3}{2})$   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 \( 1 + 0.863T + 5T^{2} \)
11 \( 1 - 2.62iT - 11T^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
17 \( 1 + 2.34iT - 17T^{2} \)
19 \( 1 - 4.09T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 - 1.57T + 29T^{2} \)
31 \( 1 - 6.29iT - 31T^{2} \)
37 \( 1 - 5.18iT - 37T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 - 0.496T + 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 4.40iT - 59T^{2} \)
61 \( 1 - 9.59iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 + 8.27T + 73T^{2} \)
79 \( 1 + 9.44iT - 79T^{2} \)
83 \( 1 - 6.62iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−8.078242503563630571797942120406, −7.28571851678463154606676292543, −7.03794160385096781636628122614, −6.11638856388923000879974991364, −5.03521019265531245900571257746, −4.73290157049701178718404843182, −3.67706499906643058600325243628, −3.04484440967910635725045144746, −1.92707584839521682211975556428, −0.896463657554551123535157626632, 0.51768619396182043250279480428, 1.64898697303894968547134158882, 2.85235815877923769658427559439, 3.40294674950971766678901633525, 4.31475533974354139115407133305, 5.16902005460139379245107281374, 5.84743102887587689220574414377, 6.47638401749633877445699455240, 7.39700216376085619001182394574, 8.031284144281494344934005676805

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