Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.126 - 0.992i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.356i·5-s + 7-s + 5.96i·11-s − 2.87i·13-s − 3.16·17-s + 0.127i·19-s − 2.80·23-s + 4.87·25-s − 2.44i·29-s + 7·31-s + 0.356i·35-s + 1.87i·37-s + 6.99·41-s + 1.12i·43-s − 7.34·47-s + ⋯
L(s)  = 1  + 0.159i·5-s + 0.377·7-s + 1.79i·11-s − 0.796i·13-s − 0.766·17-s + 0.0291i·19-s − 0.585·23-s + 0.974·25-s − 0.454i·29-s + 1.25·31-s + 0.0602i·35-s + 0.307i·37-s + 1.09·41-s + 0.171i·43-s − 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.126 - 0.992i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.126 - 0.992i)$
$L(1)$  $\approx$  $1.591184479$
$L(\frac12)$  $\approx$  $1.591184479$
$L(\frac{3}{2})$   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 \( 1 - 0.356iT - 5T^{2} \)
11 \( 1 - 5.96iT - 11T^{2} \)
13 \( 1 + 2.87iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 - 0.127iT - 19T^{2} \)
23 \( 1 + 2.80T + 23T^{2} \)
29 \( 1 + 2.44iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 - 6.99T + 41T^{2} \)
43 \( 1 - 1.12iT - 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6.63iT - 59T^{2} \)
61 \( 1 - 13.7iT - 61T^{2} \)
67 \( 1 + 8.61iT - 67T^{2} \)
71 \( 1 - 5.96T + 71T^{2} \)
73 \( 1 - 0.872T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 6.63iT - 83T^{2} \)
89 \( 1 - 6.99T + 89T^{2} \)
97 \( 1 + 9.74T + 97T^{2} \)
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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

−8.093211199314389278333853128576, −7.63314563329950704052110570434, −6.84900186573982486967947196842, −6.27551135068909372460088274197, −5.27077150608199425768713653567, −4.61117609079899404907317415697, −4.07229270836031200702358608344, −2.82424016999826177957495345324, −2.20619947455441116960769008433, −1.12485555268340234170082299514, 0.43441712241863975160851200579, 1.50629415391339233863319085576, 2.58596582219321154984449818660, 3.40061881681961208076739779359, 4.28056682474232988966333478028, 4.95115991947633861246806724770, 5.83053011585883655796712076730, 6.42652262407337434144881552150, 7.08419048170983042816042645334, 8.174377951074789952995580403480

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