Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.971 - 0.236i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.18·5-s i·7-s + 1.86i·11-s + 3.07i·13-s + 0.504i·17-s − 3.05·19-s + 5.97·23-s + 12.4·25-s + 3.52·29-s − 10.8i·31-s − 4.18i·35-s + 11.3i·37-s − 7.26i·41-s − 9.02·43-s − 0.327·47-s + ⋯
L(s)  = 1  + 1.87·5-s − 0.377i·7-s + 0.563i·11-s + 0.853i·13-s + 0.122i·17-s − 0.701·19-s + 1.24·23-s + 2.49·25-s + 0.655·29-s − 1.95i·31-s − 0.706i·35-s + 1.86i·37-s − 1.13i·41-s − 1.37·43-s − 0.0477·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.971 - 0.236i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.971 - 0.236i)$
$L(1)$  $\approx$  $3.112617805$
$L(\frac12)$  $\approx$  $3.112617805$
$L(\frac{3}{2})$   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 \( 1 - 4.18T + 5T^{2} \)
11 \( 1 - 1.86iT - 11T^{2} \)
13 \( 1 - 3.07iT - 13T^{2} \)
17 \( 1 - 0.504iT - 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + 7.26iT - 41T^{2} \)
43 \( 1 + 9.02T + 43T^{2} \)
47 \( 1 + 0.327T + 47T^{2} \)
53 \( 1 - 8.32T + 53T^{2} \)
59 \( 1 - 5.98iT - 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 - 9.21T + 67T^{2} \)
71 \( 1 - 9.02T + 71T^{2} \)
73 \( 1 - 0.416T + 73T^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 + 2.62iT - 83T^{2} \)
89 \( 1 - 5.69iT - 89T^{2} \)
97 \( 1 - 6.21T + 97T^{2} \)
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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

−8.239443040045462249272806543699, −7.12511458726586738847741949662, −6.59274063385602877638117016858, −6.13829261036567448669104768500, −5.14956527225759037644690758883, −4.71476781864738296596487460751, −3.67117278076495853068717227699, −2.48795717001459160008893575243, −1.99820312079405234402000614749, −1.03715061124041856407541926892, 0.905673221027615804166094589841, 1.85225576389688377042088369419, 2.71806382229522791413823646804, 3.29328270109801243048617175519, 4.71029071148669192441176734870, 5.36892365128502216422011027566, 5.78108961553794280806465024262, 6.59850130519712909851480512686, 7.06347037510148686041574281358, 8.380680417244081919228307005133

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