Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0841 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.46i·5-s − 7-s + 3.31i·11-s − 3.10i·13-s + 1.40·17-s + 4.80i·19-s + 8.79·23-s − 6.99·25-s − 9.87i·29-s + 7.83·31-s − 3.46i·35-s − 5.42i·37-s + 11.5·41-s + 7.03i·43-s + 11.2·47-s + ⋯
L(s)  = 1  + 1.54i·5-s − 0.377·7-s + 1.00i·11-s − 0.862i·13-s + 0.340·17-s + 1.10i·19-s + 1.83·23-s − 1.39·25-s − 1.83i·29-s + 1.40·31-s − 0.585i·35-s − 0.891i·37-s + 1.80·41-s + 1.07i·43-s + 1.64·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0841 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0841 - 0.996i)$
$L(1)$  $\approx$  $2.010118439$
$L(\frac12)$  $\approx$  $2.010118439$
$L(\frac{3}{2})$   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 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 - 4.80iT - 19T^{2} \)
23 \( 1 - 8.79T + 23T^{2} \)
29 \( 1 + 9.87iT - 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 + 5.42iT - 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 7.03iT - 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 - 3.89iT - 59T^{2} \)
61 \( 1 - 9.42iT - 61T^{2} \)
67 \( 1 + 0.909iT - 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 0.370iT - 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−7.892801525939083640281108753508, −7.56476854204422114803600451715, −6.91778622986237215403568651780, −6.11826264243551157333896108286, −5.68894081372214641686686189158, −4.48469993099034641930299915851, −3.79665341468591063869148499350, −2.72770535197689784284254344836, −2.57202557318344112564290223133, −1.01400634401365573685976491479, 0.66795814605757130207453666950, 1.24630235135724015959634406043, 2.59963805071479922986838877274, 3.43880284343233088925463534309, 4.40690065712184958411582342805, 5.01436719459181746087412059287, 5.54988613317088292615206031650, 6.54469344614984474672898993762, 7.08366484661880156994782609980, 8.089252632334886377867985666711

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