Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.126 + 0.992i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.356i·5-s + 7-s − 5.96i·11-s − 2.87i·13-s + 3.16·17-s + 0.127i·19-s + 2.80·23-s + 4.87·25-s + 2.44i·29-s + 7·31-s − 0.356i·35-s + 1.87i·37-s − 6.99·41-s + 1.12i·43-s + 7.34·47-s + ⋯
L(s)  = 1  − 0.159i·5-s + 0.377·7-s − 1.79i·11-s − 0.796i·13-s + 0.766·17-s + 0.0291i·19-s + 0.585·23-s + 0.974·25-s + 0.454i·29-s + 1.25·31-s − 0.0602i·35-s + 0.307i·37-s − 1.09·41-s + 0.171i·43-s + 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.126 + 0.992i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.126 + 0.992i)$
$L(1)$  $\approx$  $2.110672748$
$L(\frac12)$  $\approx$  $2.110672748$
$L(\frac{3}{2})$   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 \( 1 + 0.356iT - 5T^{2} \)
11 \( 1 + 5.96iT - 11T^{2} \)
13 \( 1 + 2.87iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 0.127iT - 19T^{2} \)
23 \( 1 - 2.80T + 23T^{2} \)
29 \( 1 - 2.44iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 + 6.99T + 41T^{2} \)
43 \( 1 - 1.12iT - 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6.63iT - 59T^{2} \)
61 \( 1 - 13.7iT - 61T^{2} \)
67 \( 1 + 8.61iT - 67T^{2} \)
71 \( 1 + 5.96T + 71T^{2} \)
73 \( 1 - 0.872T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + 6.99T + 89T^{2} \)
97 \( 1 + 9.74T + 97T^{2} \)
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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

−8.068286815411648599197162787173, −7.27048785628985590470788598739, −6.41505890689860611700383018600, −5.68000943845942255321817875039, −5.19058478522042885089705382405, −4.28904205375763315420395457526, −3.16567433713563689814407166149, −2.93373838658257627966009446600, −1.36825121604899471126493705249, −0.61590984346459155109150949118, 1.15590604749063937512279275809, 2.06173850842122395883349815189, 2.86609725201639784332249962472, 3.99155428195168967774439047280, 4.64116307806230046589580248023, 5.19717980471381120765343763932, 6.21168451412933263501968503588, 7.03730888196519360834995343749, 7.31019115252661511005576110907, 8.227345920436781027120124634924

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