Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.755 + 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·5-s + (−2 + 1.73i)7-s + 6i·11-s + 1.73i·13-s + 1.73·17-s + 6.92i·19-s − 3i·23-s + 6.99·25-s − 3i·29-s + 5.19i·31-s + (6.92 − 5.99i)35-s − 2·37-s − 6.92·41-s + 11·43-s − 6.92·47-s + ⋯
L(s)  = 1  − 1.54·5-s + (−0.755 + 0.654i)7-s + 1.80i·11-s + 0.480i·13-s + 0.420·17-s + 1.58i·19-s − 0.625i·23-s + 1.39·25-s − 0.557i·29-s + 0.933i·31-s + (1.17 − 1.01i)35-s − 0.328·37-s − 1.08·41-s + 1.67·43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.755 + 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.755 + 0.654i)\)
\(L(1)\)  \(\approx\)  \(0.3266624686\)
\(L(\frac12)\)  \(\approx\)  \(0.3266624686\)
\(L(\frac{3}{2})\)   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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 - 3iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.118404234428819945378087621078, −8.347088678130571822000931092976, −7.65537171919388584699562550031, −7.05116885461731810199961948161, −6.31649858095534389427346261971, −5.24491206070857153428729584453, −4.34127798906834225057164771624, −3.79108577758073422401579256900, −2.81970290182233458631684285037, −1.66021960966188214002536986596, 0.13955573195696600379917867544, 0.864449789821986888192697866052, 2.97237657547156576133852766029, 3.38197096344266498248825187001, 4.12302560734193529053718358221, 5.11427556410438191973106388713, 6.06107382667964007208527650944, 6.88820994943307952303311548771, 7.57684583726571793366998458283, 8.176095424448163332753431068417

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