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  + (2.5 + 0.866i)7-s + 1.73i·13-s − 19-s + 5·25-s + 4·31-s + 37-s + 10.3i·43-s + (5.5 + 4.33i)49-s − 8.66i·61-s + 12.1i·67-s − 1.73i·73-s + 12.1i·79-s + (−1.49 + 4.33i)91-s − 19.0i·97-s + 7·103-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)7-s + 0.480i·13-s − 0.229·19-s + 25-s + 0.718·31-s + 0.164·37-s + 1.58i·43-s + (0.785 + 0.618i)49-s − 1.10i·61-s + 1.48i·67-s − 0.202i·73-s + 1.36i·79-s + (−0.157 + 0.453i)91-s − 1.93i·97-s + 0.689·103-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  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.755 - 0.654i)\)
\(L(1)\)  \(\approx\)  \(2.052616582\)
\(L(\frac12)\)  \(\approx\)  \(2.052616582\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19.0iT - 97T^{2} \)
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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

−8.666704125815051460347817594089, −8.191198838437173175667719206220, −7.34151135788103775123796448684, −6.55611009572176393682170974254, −5.74309944355885487834802716971, −4.83376066808061143982723579449, −4.31466223597804429417569000636, −3.10106985226039559831275240406, −2.15677355281794249529549971710, −1.11333300124689030659159146722, 0.75589049633196307532403972979, 1.89869309425830551682810761125, 2.95120222167868351862903863597, 3.98672837724002051336518278271, 4.79353230927762537161910130736, 5.45569059392123083395053552887, 6.41651793704641959547709427338, 7.22706658154691271852652996124, 7.911554168247869762044599229253, 8.578426893562934369830618639324

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