Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.994 - 0.108i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.162·5-s i·7-s − 2.40i·11-s − 4.76i·13-s − 2.74i·17-s + 1.86·19-s − 1.32·23-s − 4.97·25-s − 3.96·29-s + 8.01i·31-s + 0.162i·35-s − 6.09i·37-s − 3.16i·41-s + 1.70·43-s − 12.1·47-s + ⋯
L(s)  = 1  − 0.0724·5-s − 0.377i·7-s − 0.724i·11-s − 1.32i·13-s − 0.665i·17-s + 0.426·19-s − 0.277·23-s − 0.994·25-s − 0.735·29-s + 1.43i·31-s + 0.0273i·35-s − 1.00i·37-s − 0.494i·41-s + 0.260·43-s − 1.77·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.994 - 0.108i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6048,\ (\ :1/2),\ -0.994 - 0.108i)\)
\(L(1)\)  \(\approx\)  \(0.5650731520\)
\(L(\frac12)\)  \(\approx\)  \(0.5650731520\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.162T + 5T^{2} \)
11 \( 1 + 2.40iT - 11T^{2} \)
13 \( 1 + 4.76iT - 13T^{2} \)
17 \( 1 + 2.74iT - 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
23 \( 1 + 1.32T + 23T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 - 8.01iT - 31T^{2} \)
37 \( 1 + 6.09iT - 37T^{2} \)
41 \( 1 + 3.16iT - 41T^{2} \)
43 \( 1 - 1.70T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 8.22T + 53T^{2} \)
59 \( 1 + 5.45iT - 59T^{2} \)
61 \( 1 + 4.28iT - 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 6.37T + 73T^{2} \)
79 \( 1 - 9.75iT - 79T^{2} \)
83 \( 1 - 16.4iT - 83T^{2} \)
89 \( 1 - 2.35iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−7.70807826885304009556524918736, −7.13524015399172033717695680868, −6.23657423966478219592796078047, −5.48772629901710437624762131248, −5.00340687002863745401813787983, −3.81059995086622059276105095622, −3.34100687572114034031108300446, −2.39346105864594494281275289366, −1.15892986908857978587009040857, −0.14572212169402424235894592357, 1.58445878275771928526532611161, 2.15408847156672865822450928861, 3.27535600568013267637577697051, 4.20565085947978584011092499310, 4.64500930872610302849800091837, 5.76819025108907843618684544379, 6.20438389078143442888373793075, 7.12467812198161735987913546695, 7.65142297663997960629279508377, 8.456657106822548441806062232282

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