Properties

Label 2-6048-12.11-c1-0-11
Degree $2$
Conductor $6048$
Sign $0.707 - 0.707i$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8015096804\)
\(L(\frac12)\) \(\approx\) \(0.8015096804\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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