Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.587 - 0.809i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.63i·5-s + 7-s − 4.29i·11-s − 0.0480i·13-s + 3.16·17-s + 0.726i·19-s − 5.28·23-s − 1.94·25-s + 8.00i·29-s − 4.90·31-s + 2.63i·35-s + 9.10i·37-s − 1.45·41-s − 8.85i·43-s − 9.23·47-s + ⋯
L(s)  = 1  + 1.17i·5-s + 0.377·7-s − 1.29i·11-s − 0.0133i·13-s + 0.766·17-s + 0.166i·19-s − 1.10·23-s − 0.388·25-s + 1.48i·29-s − 0.881·31-s + 0.445i·35-s + 1.49i·37-s − 0.227·41-s − 1.35i·43-s − 1.34·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.587 - 0.809i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.587 - 0.809i)$
$L(1)$  $\approx$  $1.320386908$
$L(\frac12)$  $\approx$  $1.320386908$
$L(\frac{3}{2})$   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 \( 1 - 2.63iT - 5T^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
13 \( 1 + 0.0480iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 0.726iT - 19T^{2} \)
23 \( 1 + 5.28T + 23T^{2} \)
29 \( 1 - 8.00iT - 29T^{2} \)
31 \( 1 + 4.90T + 31T^{2} \)
37 \( 1 - 9.10iT - 37T^{2} \)
41 \( 1 + 1.45T + 41T^{2} \)
43 \( 1 + 8.85iT - 43T^{2} \)
47 \( 1 + 9.23T + 47T^{2} \)
53 \( 1 - 3.14iT - 53T^{2} \)
59 \( 1 - 5.90iT - 59T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 - 8.55iT - 67T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + 7.56T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 9.05iT - 83T^{2} \)
89 \( 1 + 5.15T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.328893380298959425172523047336, −7.55239310529290695861714318713, −6.94228195472977108140096190366, −6.17492527449638971786375457731, −5.61861644267383883158577897390, −4.78226154794527299546290543008, −3.48688174518002179011681708830, −3.36276281660016150273733665341, −2.27106449245938900064045305917, −1.19383714320830173237702930981, 0.34072835190528893378823779008, 1.57820048152324375141538590388, 2.17279197961991589968550523824, 3.50205666043321454314829952789, 4.39898629123723135991886196665, 4.80352071398497189227832450305, 5.59961096177678184546818168865, 6.30166945846588969112820665004, 7.35853533140055063477488365705, 7.83579883963515812775714554199

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