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  =  $(2,\ 6048,\ (\ :1/2),\ 0.997 - 0.0726i)$
$L(1)$  $\approx$  $1.107814304$
$L(\frac12)$  $\approx$  $1.107814304$
$L(\frac{3}{2})$   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

−7.987784933879810082990095120595, −7.32745272703215099363542540133, −6.96756473725090573371080699137, −6.01088927410428222362172073178, −4.95708247002924207533866350488, −4.46223848046850841124479861526, −3.66853771985040648596779928148, −3.00285964316588691230909348116, −1.79870857970520998743530053372, −0.57181651928980280830723857314, 0.52392351764112644112895090051, 1.80693300165186365255147772401, 2.99387432664040430268337653643, 3.63155787217996986449724919924, 4.28689960659680930828877562934, 5.21790028786421809883680490349, 5.90279048209183522253065203455, 6.69366254580728825153051554934, 7.61037630095099738338556282994, 7.949390614157186415232639655655

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