Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.87·5-s + (2 + 1.73i)7-s + 2.23i·11-s + 3.46i·13-s − 5.19i·19-s + 2.23i·23-s + 10.0·25-s + 4.47i·29-s + 1.73i·31-s + (7.74 + 6.70i)35-s − 37-s + 3.87·41-s − 2·43-s − 7.74·47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  + 1.73·5-s + (0.755 + 0.654i)7-s + 0.674i·11-s + 0.960i·13-s − 1.19i·19-s + 0.466i·23-s + 2.00·25-s + 0.830i·29-s + 0.311i·31-s + (1.30 + 1.13i)35-s − 0.164·37-s + 0.604·41-s − 0.304·43-s − 1.12·47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.654 - 0.755i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.654 - 0.755i)\)
\(L(1)\)  \(\approx\)  \(2.880731239\)
\(L(\frac12)\)  \(\approx\)  \(2.880731239\)
\(L(\frac{3}{2})\)   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 \( 1 - 3.87T + 5T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 2.23iT - 23T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 3.87T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 + 8.94iT - 53T^{2} \)
59 \( 1 - 7.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−8.970294632130577726678951333616, −8.294242166855991558384206552025, −7.01884987463605748443026353290, −6.66948538371421514004885982389, −5.60391034936522234635863257996, −5.15271720385637278652934674306, −4.37765547575386927378736418760, −2.88614031195433916707824735795, −2.06152133928219905593189017382, −1.47473377615761563774349049437, 0.930789214913597015010690889754, 1.85931143963676880871271953232, 2.79017399054597350358773250514, 3.87974922662895039700121734281, 4.94093571781666379173901085359, 5.67649203532205639063722907319, 6.12620587509765358537298960920, 7.05183964054449592016492604359, 8.076978817456478406480305257709, 8.459602225507776480902592475863

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