Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.997 - 0.0726i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.90·5-s i·7-s − 1.28i·11-s + 1.86i·13-s + 3.87i·17-s + 2.04·19-s + 0.934·23-s + 3.45·25-s + 2.98·29-s + 1.85i·31-s − 2.90i·35-s − 3.85i·37-s − 8.72i·41-s + 1.19·43-s + 11.1·47-s + ⋯
L(s)  = 1  + 1.30·5-s − 0.377i·7-s − 0.388i·11-s + 0.518i·13-s + 0.939i·17-s + 0.468·19-s + 0.194·23-s + 0.691·25-s + 0.553·29-s + 0.333i·31-s − 0.491i·35-s − 0.634i·37-s − 1.36i·41-s + 0.181·43-s + 1.62·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.997 - 0.0726i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.997 - 0.0726i)$
$L(1)$  $\approx$  $2.743138743$
$L(\frac12)$  $\approx$  $2.743138743$
$L(\frac{3}{2})$   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 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 - 1.86iT - 13T^{2} \)
17 \( 1 - 3.87iT - 17T^{2} \)
19 \( 1 - 2.04T + 19T^{2} \)
23 \( 1 - 0.934T + 23T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 - 1.85iT - 31T^{2} \)
37 \( 1 + 3.85iT - 37T^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 - 1.19T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 - 4.70iT - 61T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 - 9.72T + 71T^{2} \)
73 \( 1 - 9.40T + 73T^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 - 3.72iT - 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−8.168524578777860747595172646076, −7.21361850389308723864662966502, −6.66998450663107373880171848041, −5.79683596507269933627650659445, −5.50324070671641611127100146932, −4.41995069474313439754702016037, −3.69048329702704084171374355844, −2.64304732374113656447377092895, −1.86405854901708987435443960316, −0.941028894050565138960152363308, 0.855583516188966316940702149403, 1.94562784674886938832036061289, 2.65067035044115235795963660064, 3.43452334477851557937140476042, 4.79142468368845193630252276731, 5.10387293716500272268467175088, 6.04655500021998050743474809982, 6.42588007776293202188330407750, 7.39394982464681380197232107619, 7.999993805666755006730945091222

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