Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.940 - 0.340i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.436·5-s i·7-s + 2.73i·11-s − 1.64i·13-s + 5.22i·17-s + 5.99·19-s + 7.68·23-s − 4.80·25-s − 4.94·29-s − 9.77i·31-s − 0.436i·35-s + 3.81i·37-s − 9.74i·41-s + 8.84·43-s − 4.54·47-s + ⋯
L(s)  = 1  + 0.195·5-s − 0.377i·7-s + 0.825i·11-s − 0.456i·13-s + 1.26i·17-s + 1.37·19-s + 1.60·23-s − 0.961·25-s − 0.918·29-s − 1.75i·31-s − 0.0737i·35-s + 0.627i·37-s − 1.52i·41-s + 1.34·43-s − 0.663·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.940 - 0.340i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.940 - 0.340i)$
$L(1)$  $\approx$  $2.107170850$
$L(\frac12)$  $\approx$  $2.107170850$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 - 0.436T + 5T^{2} \)
11 \( 1 - 2.73iT - 11T^{2} \)
13 \( 1 + 1.64iT - 13T^{2} \)
17 \( 1 - 5.22iT - 17T^{2} \)
19 \( 1 - 5.99T + 19T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + 9.77iT - 31T^{2} \)
37 \( 1 - 3.81iT - 37T^{2} \)
41 \( 1 + 9.74iT - 41T^{2} \)
43 \( 1 - 8.84T + 43T^{2} \)
47 \( 1 + 4.54T + 47T^{2} \)
53 \( 1 - 9.94T + 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 - 6.26iT - 61T^{2} \)
67 \( 1 + 9.42T + 67T^{2} \)
71 \( 1 - 9.76T + 71T^{2} \)
73 \( 1 + 9.14T + 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 + 0.227iT - 83T^{2} \)
89 \( 1 + 2.46iT - 89T^{2} \)
97 \( 1 + 3.37T + 97T^{2} \)
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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

−7.87051434836180027096171950729, −7.48673597757959707070401869004, −6.84146348638702351389770368544, −5.75572603230083455431524838723, −5.47826602539408827470969620665, −4.34276888530636408271586430986, −3.80565044530676261912975792430, −2.80822952049659278029679260557, −1.87041144849331790268025760099, −0.870247692780859077417944506116, 0.70408771009388275660510121273, 1.74780007529477644136704064834, 2.95083586612624418448994668246, 3.31027696185316886867196140215, 4.52899264278391338747421663697, 5.28905724314204069723139212266, 5.72883402972383824679458685309, 6.73907136937075130470651786970, 7.25421830752026719779043813325, 8.010696382714398018788928964733

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