Properties

Degree $2$
Conductor $6048$
Sign $0.972 + 0.232i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.512i·5-s + 7-s − 1.82i·11-s + 1.80i·13-s + 8.11·17-s − 3.43i·19-s + 3.65·23-s + 4.73·25-s + 7.98i·29-s + 1.56·31-s − 0.512i·35-s − 8.44i·37-s − 2.30·41-s + 10.7i·43-s − 11.3·47-s + ⋯
L(s)  = 1  − 0.229i·5-s + 0.377·7-s − 0.551i·11-s + 0.500i·13-s + 1.96·17-s − 0.789i·19-s + 0.762·23-s + 0.947·25-s + 1.48i·29-s + 0.281·31-s − 0.0866i·35-s − 1.38i·37-s − 0.360·41-s + 1.64i·43-s − 1.66·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.972 + 0.232i$
Motivic weight: \(1\)
Character: $\chi_{6048} (3025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ 0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.350613405\)
\(L(\frac12)\) \(\approx\) \(2.350613405\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 + 0.512iT - 5T^{2} \)
11 \( 1 + 1.82iT - 11T^{2} \)
13 \( 1 - 1.80iT - 13T^{2} \)
17 \( 1 - 8.11T + 17T^{2} \)
19 \( 1 + 3.43iT - 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + 8.44iT - 37T^{2} \)
41 \( 1 + 2.30T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 8.87iT - 53T^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 + 5.31iT - 61T^{2} \)
67 \( 1 + 6.44iT - 67T^{2} \)
71 \( 1 + 9.10T + 71T^{2} \)
73 \( 1 - 9.24T + 73T^{2} \)
79 \( 1 + 3.64T + 79T^{2} \)
83 \( 1 + 4.53iT - 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−8.025300087664493246434516859732, −7.39479067917384702850320224549, −6.67959839939323695043212269914, −5.85586148765416215661547498599, −5.08078516642043890519745774245, −4.62645059647492301993954670515, −3.40236593810240085740679734168, −2.97195998254855969166175897416, −1.61999138797086821944552135420, −0.840687105754759906877950431285, 0.871504363753324296784721997554, 1.81708341931164581213910328660, 2.94448828955579518180830320912, 3.53157049060258855553943722829, 4.55465142856744899979477210159, 5.26687946738880879290961469608, 5.87420951674416922121556923500, 6.78138104871971529034767514357, 7.43084097548379570597035172095, 8.120913175818675964070806482848

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