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  − 4.38i·5-s i·7-s + 2.31·11-s − 4.73·13-s − 5.27i·17-s − 5i·19-s + 0.517·23-s − 14.1·25-s + 4.24i·29-s − 3.53i·31-s − 4.38·35-s − 4.46·37-s + 6.45i·41-s − 1.26i·43-s − 8.10·47-s + ⋯
L(s)  = 1  − 1.95i·5-s − 0.377i·7-s + 0.696·11-s − 1.31·13-s − 1.28i·17-s − 1.14i·19-s + 0.107·23-s − 2.83·25-s + 0.787i·29-s − 0.635i·31-s − 0.740·35-s − 0.733·37-s + 1.00i·41-s − 0.193i·43-s − 1.18·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.8548949013$
$L(\frac12)$  $\approx$  $0.8548949013$
$L(\frac{3}{2})$   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 \( 1 + 4.38iT - 5T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 - 0.517T + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 + 3.53iT - 31T^{2} \)
37 \( 1 + 4.46T + 37T^{2} \)
41 \( 1 - 6.45iT - 41T^{2} \)
43 \( 1 + 1.26iT - 43T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 - 5.93iT - 53T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 - 9.66iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + 0.517iT - 89T^{2} \)
97 \( 1 - 0.928T + 97T^{2} \)
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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

−7.56247757769646554023489124840, −7.13670424764229698127630586413, −6.14570334585667639954395787424, −5.09211252208485966549059008505, −4.87121533097530527517568815549, −4.25596404548046373838049239114, −3.13973349923110398369655981234, −2.02797123380426936886867699404, −1.03518939179099225735620068217, −0.22948232768961043615105816017, 1.82855553907130628820850649175, 2.38337746568990344065629523210, 3.44749787156701084663076288907, 3.77639046749902164265214279759, 4.97795678840451598657634167899, 5.94695473769334744502876088714, 6.42503676233018746864611147775, 7.01634478329074750255753367130, 7.73084395134079075591686323295, 8.301707632003993182309709566239

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