Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.826 + 0.563i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.52i·5-s − 7-s + 5.70i·11-s + 3.06i·13-s − 5.49·17-s + 7.28i·19-s − 0.539·23-s − 1.38·25-s − 8.35i·29-s − 6.74·31-s − 2.52i·35-s + 10.2i·37-s + 5.58·41-s + 3.98i·43-s − 4.83·47-s + ⋯
L(s)  = 1  + 1.12i·5-s − 0.377·7-s + 1.71i·11-s + 0.848i·13-s − 1.33·17-s + 1.67i·19-s − 0.112·23-s − 0.276·25-s − 1.55i·29-s − 1.21·31-s − 0.427i·35-s + 1.68i·37-s + 0.872·41-s + 0.607i·43-s − 0.705·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.826 + 0.563i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.826 + 0.563i)$
$L(1)$  $\approx$  $0.9046843633$
$L(\frac12)$  $\approx$  $0.9046843633$
$L(\frac{3}{2})$   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 \( 1 - 2.52iT - 5T^{2} \)
11 \( 1 - 5.70iT - 11T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 + 5.49T + 17T^{2} \)
19 \( 1 - 7.28iT - 19T^{2} \)
23 \( 1 + 0.539T + 23T^{2} \)
29 \( 1 + 8.35iT - 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 - 3.98iT - 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 - 10.1iT - 59T^{2} \)
61 \( 1 - 4.14iT - 61T^{2} \)
67 \( 1 + 5.96iT - 67T^{2} \)
71 \( 1 - 4.93T + 71T^{2} \)
73 \( 1 - 8.66T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 5.83iT - 83T^{2} \)
89 \( 1 - 9.28T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.400510041153483937116683419090, −7.61085091631226768155156237099, −7.00417708455286046845743024809, −6.50578345092676767609096928707, −5.89444831002558100862313839490, −4.66650413071438039748119202421, −4.17225811552678532364426657595, −3.30484355422941595711742436320, −2.26452577023267493725061747598, −1.79002229191483692473834647574, 0.26331861170681083869672527679, 0.920788483595561710583194710915, 2.28471906202564851448548023290, 3.19003437145482329086385788313, 3.93333572331787940377327364558, 4.94495160353685830388634687501, 5.38169003694391935724098759402, 6.16373734460519624817197589826, 6.92390117893546530271851248385, 7.73115987410144979345289144388

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