Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.992 + 0.123i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.26i·5-s − 7-s − 2.02i·11-s + 4.05i·13-s + 1.33·17-s − 3.28i·19-s + 2.25·23-s + 3.38·25-s + 3.58i·29-s + 0.464·31-s + 1.26i·35-s + 11.8i·37-s + 1.73·41-s − 1.70i·43-s − 8.45·47-s + ⋯
L(s)  = 1  − 0.567i·5-s − 0.377·7-s − 0.611i·11-s + 1.12i·13-s + 0.324·17-s − 0.753i·19-s + 0.470·23-s + 0.677·25-s + 0.666i·29-s + 0.0833·31-s + 0.214i·35-s + 1.95i·37-s + 0.271·41-s − 0.259i·43-s − 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.992 + 0.123i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.992 + 0.123i)$
$L(1)$  $\approx$  $1.801741693$
$L(\frac12)$  $\approx$  $1.801741693$
$L(\frac{3}{2})$   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 \( 1 + 1.26iT - 5T^{2} \)
11 \( 1 + 2.02iT - 11T^{2} \)
13 \( 1 - 4.05iT - 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 + 3.28iT - 19T^{2} \)
23 \( 1 - 2.25T + 23T^{2} \)
29 \( 1 - 3.58iT - 29T^{2} \)
31 \( 1 - 0.464T + 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 + 1.70iT - 43T^{2} \)
47 \( 1 + 8.45T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 - 5.49iT - 59T^{2} \)
61 \( 1 - 6.02iT - 61T^{2} \)
67 \( 1 + 0.381iT - 67T^{2} \)
71 \( 1 - 9.47T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 4.90T + 79T^{2} \)
83 \( 1 + 6.95iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 7.62T + 97T^{2} \)
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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

−8.267086169592225506536311491575, −7.20210518888119625541306351697, −6.69212845166960504958420793725, −6.00139913116821160706202688605, −4.99863247631526968331147004359, −4.62892875201953423689140927650, −3.53728183312394184369897662832, −2.88616762092352648319198303500, −1.71339120566926478371512380018, −0.74549912146161814047313822636, 0.67649630745103822945478978075, 1.96763904103143338714472020810, 2.90160950109082403316791182236, 3.50563469630401526207440331461, 4.42619206548088904685200549069, 5.31903053284964491769340373121, 5.98437946594622493252126960524, 6.66950807768554629592977383971, 7.49564601429359637389366537276, 7.86264039068763223801772456213

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