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  =  \((2,\ 3024,\ (\ :1/2),\ 0.755 + 0.654i)\)
\(L(1)\)  \(\approx\)  \(2.200743948\)
\(L(\frac12)\)  \(\approx\)  \(2.200743948\)
\(L(\frac{3}{2})\)   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

−8.978036484082466627159876044056, −7.69064741107763508893253208839, −6.98405445273500992193139130800, −6.47259835803484184842531354413, −5.63882476350076115983387494833, −4.76193153964368895152295759134, −4.05983552332924558221594504943, −2.56708568040220006853202737918, −2.21641373888780955690045618647, −0.74721693379687147814401437040, 1.15342635784500133827281101525, 2.23905039686556403715569207046, 3.05919087698111679810180064815, 3.93785290566110652860033438457, 5.39318161614399163435073121046, 5.84419731698912348304469408878, 6.20480968573751853862534011159, 7.14429514582428275444996754877, 8.305720102095961336309075484892, 9.038633713064029251362733574821

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