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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $0.8015096804$
$L(\frac12)$  $\approx$  $0.8015096804$
$L(\frac{3}{2})$   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

−7.74674986162233918432572059656, −7.18229024209235176679797702754, −6.78892941122885752285652058517, −5.96067776048972145569267458985, −5.22258753322270790206905781927, −4.12305299684380724796569864508, −3.51617740596616161805759118630, −2.44800274358054743428837571586, −2.25268116770506600838995982866, −0.24150242015579209210053678799, 0.855021973796694489965699293567, 1.87873516534730473316562516984, 2.77126538799763323180015285737, 3.93113120071195189445366304811, 4.79570545187606676258925098186, 5.22834393307799241497571772510, 5.63864538844424297627866005233, 6.85251204334669031459795056318, 7.65305740332449679102422518616, 8.215692611129798149389617732983

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