Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.951 + 0.309i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0549i·5-s + 7-s − 2.63i·11-s + 3.67i·13-s + 3.16·17-s + 3.07i·19-s − 2.86·23-s + 4.99·25-s − 10.1i·29-s − 9.32·31-s − 0.0549i·35-s − 0.774i·37-s + 6.36·41-s − 9.98i·43-s + 12.3·47-s + ⋯
L(s)  = 1  − 0.0245i·5-s + 0.377·7-s − 0.794i·11-s + 1.02i·13-s + 0.766·17-s + 0.706i·19-s − 0.598·23-s + 0.999·25-s − 1.89i·29-s − 1.67·31-s − 0.00928i·35-s − 0.127i·37-s + 0.993·41-s − 1.52i·43-s + 1.79·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.951 + 0.309i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.951 + 0.309i)$
$L(1)$  $\approx$  $2.068488358$
$L(\frac12)$  $\approx$  $2.068488358$
$L(\frac{3}{2})$   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 \( 1 + 0.0549iT - 5T^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
13 \( 1 - 3.67iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 3.07iT - 19T^{2} \)
23 \( 1 + 2.86T + 23T^{2} \)
29 \( 1 + 10.1iT - 29T^{2} \)
31 \( 1 + 9.32T + 31T^{2} \)
37 \( 1 + 0.774iT - 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 9.98iT - 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 3.39iT - 53T^{2} \)
59 \( 1 - 6.93iT - 59T^{2} \)
61 \( 1 - 8.35iT - 61T^{2} \)
67 \( 1 + 8.93iT - 67T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 - 8.38T + 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.88793376639341913919516870888, −7.51539727390871856547119073870, −6.58043700552106801229079306927, −5.80477088227986554607639883214, −5.36276687473438632305588958585, −4.15373276572694499707224831059, −3.84199785565798726485717577308, −2.65453025978903839783036342956, −1.81503362218473476574393463283, −0.70193043259938654259965434906, 0.845744576586133467966226848296, 1.86897003357184962558569056632, 2.87682648289940124807415105767, 3.59821093002769735986803301044, 4.62306014111878173647671380979, 5.21025342597636660080024956271, 5.85286326102388594930165004734, 6.86979139347548948978285204421, 7.41212731214362068471524156422, 8.000559851532454021296795339395

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