Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.936 - 0.351i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.25i·5-s − 7-s + 1.55i·11-s + 1.07i·13-s + 0.0158·17-s − 2.35i·19-s − 5.95·23-s + 3.42·25-s − 0.469i·29-s − 1.69·31-s + 1.25i·35-s + 4.59i·37-s + 12.2·41-s + 1.97i·43-s − 7.12·47-s + ⋯
L(s)  = 1  − 0.560i·5-s − 0.377·7-s + 0.469i·11-s + 0.297i·13-s + 0.00383·17-s − 0.539i·19-s − 1.24·23-s + 0.685·25-s − 0.0872i·29-s − 0.303·31-s + 0.211i·35-s + 0.755i·37-s + 1.90·41-s + 0.301i·43-s − 1.03·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.936 - 0.351i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.936 - 0.351i)$
$L(1)$  $\approx$  $1.585884903$
$L(\frac12)$  $\approx$  $1.585884903$
$L(\frac{3}{2})$   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 \( 1 + 1.25iT - 5T^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 - 1.07iT - 13T^{2} \)
17 \( 1 - 0.0158T + 17T^{2} \)
19 \( 1 + 2.35iT - 19T^{2} \)
23 \( 1 + 5.95T + 23T^{2} \)
29 \( 1 + 0.469iT - 29T^{2} \)
31 \( 1 + 1.69T + 31T^{2} \)
37 \( 1 - 4.59iT - 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + 1.86iT - 53T^{2} \)
59 \( 1 - 8.54iT - 59T^{2} \)
61 \( 1 - 3.92iT - 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 4.22T + 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 - 8.88iT - 83T^{2} \)
89 \( 1 + 0.240T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−8.094114845184330965578205647811, −7.46756810575629521754609737257, −6.63156620123038598940191586262, −6.07219235879352040635211272554, −5.14692891307596475933523620144, −4.52010913664180774242837327408, −3.79040663820346889234679305572, −2.78251162105374335657476396172, −1.90334958089198800586402570733, −0.78257062917175532230425216028, 0.54717505538327512160774077216, 1.88985947189456218623634505761, 2.80475480690486511147202624691, 3.54723636884727902870631161255, 4.24518400040155567325758141273, 5.31287230817917842259618912834, 5.98349751842768371808646977887, 6.54242959974203410642469557351, 7.36121483002533524466189607240, 7.976886603280425063166055539970

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