Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.944 - 0.327i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·5-s + (−2 − 1.73i)7-s + 3.46i·11-s − 3.46i·13-s − 5.19i·17-s + 2·19-s + 6.92i·23-s + 2.00·25-s − 6·29-s + 10·31-s + (2.99 − 3.46i)35-s − 11·37-s − 1.73i·41-s − 5.19i·43-s + 9·47-s + ⋯
L(s)  = 1  + 0.774i·5-s + (−0.755 − 0.654i)7-s + 1.04i·11-s − 0.960i·13-s − 1.26i·17-s + 0.458·19-s + 1.44i·23-s + 0.400·25-s − 1.11·29-s + 1.79·31-s + (0.507 − 0.585i)35-s − 1.80·37-s − 0.270i·41-s − 0.792i·43-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.944 - 0.327i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.944 - 0.327i)\)
\(L(1)\)  \(\approx\)  \(1.569335425\)
\(L(\frac12)\)  \(\approx\)  \(1.569335425\)
\(L(\frac{3}{2})\)   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 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 + 5.19iT - 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−8.889803006369873202301972598927, −7.69458897784687602651183754310, −7.19066119438422454931155318321, −6.80649704633291353046573366105, −5.66405210675233930591190299267, −4.99663902628604351174422545490, −3.83005413877166323974061444072, −3.18705157859581624604303721240, −2.30506852419882470703272665207, −0.793600136377040141638880833366, 0.72386858374236196768303796602, 2.00099562698686908988470418507, 3.06174565463975225853253250619, 3.95732015033460497752393201286, 4.81274265967626635064188154403, 5.76212709247606018635998308512, 6.28140314945068241546060607494, 7.05694132376657547494061318262, 8.275315006751149221921348984980, 8.710497640101815835512361123389

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