Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.972 - 0.232i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.972 - 0.232i)$
$L(1)$  $\approx$  $2.350613405$
$L(\frac12)$  $\approx$  $2.350613405$
$L(\frac{3}{2})$   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 \( 1 - T \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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