Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.955 - 0.295i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.04i·5-s − 7-s + 0.128i·11-s − 6.30i·13-s − 5.32·17-s − 6.68i·19-s + 5.18·23-s − 4.26·25-s − 9.96i·29-s − 3.27·31-s + 3.04i·35-s + 0.796i·37-s + 2.96·41-s + 6.99i·43-s + 4.76·47-s + ⋯
L(s)  = 1  − 1.36i·5-s − 0.377·7-s + 0.0387i·11-s − 1.74i·13-s − 1.29·17-s − 1.53i·19-s + 1.08·23-s − 0.852·25-s − 1.85i·29-s − 0.587·31-s + 0.514i·35-s + 0.131i·37-s + 0.463·41-s + 1.06i·43-s + 0.695·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.955 - 0.295i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.955 - 0.295i)$
$L(1)$  $\approx$  $1.091523810$
$L(\frac12)$  $\approx$  $1.091523810$
$L(\frac{3}{2})$   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 \( 1 + 3.04iT - 5T^{2} \)
11 \( 1 - 0.128iT - 11T^{2} \)
13 \( 1 + 6.30iT - 13T^{2} \)
17 \( 1 + 5.32T + 17T^{2} \)
19 \( 1 + 6.68iT - 19T^{2} \)
23 \( 1 - 5.18T + 23T^{2} \)
29 \( 1 + 9.96iT - 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 - 0.796iT - 37T^{2} \)
41 \( 1 - 2.96T + 41T^{2} \)
43 \( 1 - 6.99iT - 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 - 14.6iT - 61T^{2} \)
67 \( 1 + 1.05iT - 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 - 6.46iT - 83T^{2} \)
89 \( 1 - 2.23T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−7.75464404258991322480169714308, −7.06201356105968381518566663433, −6.13725806977147538011873700452, −5.48085549060110504281652003736, −4.74856015176793639634899422199, −4.26054308084976791753433724089, −3.06896357343905483518194259338, −2.37257768263168332964327268789, −0.979782502952307566260355828566, −0.31358697821127752662686245440, 1.60867051783111031819396828291, 2.37415810661989466011300271931, 3.31561594984687573011643179102, 3.91244704681367386686077522789, 4.77955605080916755304073048911, 5.83088143674088565906692721700, 6.49499810067950892467514469282, 7.06477099822570499627170442992, 7.36664593920476428546200040489, 8.680094643204857537677489182934

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