Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.258 + 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.12·5-s + i·7-s + 3.86i·11-s − 5.08i·13-s − 2i·17-s − 7.99·19-s + 7.68·23-s − 3.72·25-s − 4.73·29-s − 4.08i·31-s + 1.12i·35-s + 8.28i·37-s − 8.41i·41-s + 10.0·43-s + 1.93·47-s + ⋯
L(s)  = 1  + 0.505·5-s + 0.377i·7-s + 1.16i·11-s − 1.41i·13-s − 0.485i·17-s − 1.83·19-s + 1.60·23-s − 0.744·25-s − 0.878·29-s − 0.733i·31-s + 0.190i·35-s + 1.36i·37-s − 1.31i·41-s + 1.52·43-s + 0.282·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.258 + 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.258 + 0.965i)$
$L(1)$  $\approx$  $1.141134958$
$L(\frac12)$  $\approx$  $1.141134958$
$L(\frac{3}{2})$   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 \( 1 - iT \)
good5 \( 1 - 1.12T + 5T^{2} \)
11 \( 1 - 3.86iT - 11T^{2} \)
13 \( 1 + 5.08iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 7.99T + 19T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 4.08iT - 31T^{2} \)
37 \( 1 - 8.28iT - 37T^{2} \)
41 \( 1 + 8.41iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + 5.08T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 2.63T + 73T^{2} \)
79 \( 1 - 7.17iT - 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 - 4.17T + 97T^{2} \)
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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

−7.78368519456732917254358345511, −7.24238450665304613515229018074, −6.36470116558301642591383049510, −5.75624261470818134723116121765, −4.99591756601103020420630964643, −4.36001920487202284297388321824, −3.27474327990519760450708067833, −2.43282578213911629451118998684, −1.72927384573438489140861982405, −0.28270229787612114910157787252, 1.18055839595837711451397901288, 2.07593939814274949919371839345, 3.00063672902453862189961085630, 4.05251149074036688883778345444, 4.45246326476399808114026527561, 5.64788500434882206579365602749, 6.10080107723062865086977352524, 6.82147749565153608989403108918, 7.47969148793986006900093304191, 8.435573547035836797754738137330

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