Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.276 - 0.961i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82i·5-s − 7-s − 3.75i·11-s − 3.09i·13-s − 3.61·17-s + 1.65i·19-s − 7.47·23-s + 1.65·25-s − 2.38i·29-s − 8.97·31-s + 1.82i·35-s + 8.94i·37-s − 3.04·41-s + 1.82i·43-s + 6.21·47-s + ⋯
L(s)  = 1  − 0.818i·5-s − 0.377·7-s − 1.13i·11-s − 0.859i·13-s − 0.875·17-s + 0.379i·19-s − 1.55·23-s + 0.330·25-s − 0.443i·29-s − 1.61·31-s + 0.309i·35-s + 1.47i·37-s − 0.475·41-s + 0.278i·43-s + 0.906·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.276 - 0.961i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.276 - 0.961i)$
$L(1)$  $\approx$  $0.1637269423$
$L(\frac12)$  $\approx$  $0.1637269423$
$L(\frac{3}{2})$   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 \( 1 + 1.82iT - 5T^{2} \)
11 \( 1 + 3.75iT - 11T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 - 1.65iT - 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 + 2.38iT - 29T^{2} \)
31 \( 1 + 8.97T + 31T^{2} \)
37 \( 1 - 8.94iT - 37T^{2} \)
41 \( 1 + 3.04T + 41T^{2} \)
43 \( 1 - 1.82iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 + 3.21iT - 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 + 9.91iT - 61T^{2} \)
67 \( 1 - 8.73iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 7.24T + 79T^{2} \)
83 \( 1 - 8.99iT - 83T^{2} \)
89 \( 1 - 1.54T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.378125054376851318824153105038, −7.74854845140464639224088249487, −6.81077997966603965701405296907, −5.97567472392553498709400700039, −5.57556667725083032682890619922, −4.67517471221604967166401976857, −3.85396102786203559086285858067, −3.14031449237539099458468686472, −2.10153533188236817423854123087, −0.962064392076475824609611251743, 0.04537146602618752952059079181, 1.89246898671864289578532199786, 2.31407876473810550175620715479, 3.47983738469773811785426830150, 4.13466379013916661083364386503, 4.89307806059617871229938063361, 5.88851805438842068088928240961, 6.56282845345648138838219786480, 7.18871925898597890141384698321, 7.54352668536440832545577281283

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