Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.826 - 0.563i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.52i·5-s − 7-s − 5.70i·11-s + 3.06i·13-s + 5.49·17-s + 7.28i·19-s + 0.539·23-s − 1.38·25-s + 8.35i·29-s − 6.74·31-s + 2.52i·35-s + 10.2i·37-s − 5.58·41-s + 3.98i·43-s + 4.83·47-s + ⋯
L(s)  = 1  − 1.12i·5-s − 0.377·7-s − 1.71i·11-s + 0.848i·13-s + 1.33·17-s + 1.67i·19-s + 0.112·23-s − 0.276·25-s + 1.55i·29-s − 1.21·31-s + 0.427i·35-s + 1.68i·37-s − 0.872·41-s + 0.607i·43-s + 0.705·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.826 - 0.563i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.826 - 0.563i)$
$L(1)$  $\approx$  $1.516464153$
$L(\frac12)$  $\approx$  $1.516464153$
$L(\frac{3}{2})$   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 \( 1 + 2.52iT - 5T^{2} \)
11 \( 1 + 5.70iT - 11T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 - 5.49T + 17T^{2} \)
19 \( 1 - 7.28iT - 19T^{2} \)
23 \( 1 - 0.539T + 23T^{2} \)
29 \( 1 - 8.35iT - 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + 5.58T + 41T^{2} \)
43 \( 1 - 3.98iT - 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 - 4.14iT - 61T^{2} \)
67 \( 1 + 5.96iT - 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 - 8.66T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 5.83iT - 83T^{2} \)
89 \( 1 + 9.28T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.306996214517805362167609838171, −7.58477099085768736622942250885, −6.62655387647608498711922819375, −5.82850833331372921753789458913, −5.43297954673078668075550185149, −4.56451931049401484866674843430, −3.53409010237672926878073241785, −3.20166362894334016700763578369, −1.60261862329206593572192102879, −1.00884246225050178491521901746, 0.44197698104252919093145469429, 2.00463264809297039919460520844, 2.68715171124867381224553322027, 3.45688891452991902737456556585, 4.28915882766614765970098036795, 5.23381938785630537591751684229, 5.85575035759980581407251579116, 6.85487355124074903773365994512, 7.25318655589556196804863220682, 7.66749769681163317033624574066

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