Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.933 - 0.357i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·5-s + (0.946 + 2.47i)7-s + 0.464i·11-s − 3.03i·13-s − 0.900·17-s − 0.831i·19-s − 7.12i·23-s + 2.33·25-s + 4.22i·29-s − 1.30i·31-s + (−2.56 − 6.69i)35-s + 10.3·37-s − 5.51·41-s + 11.8·43-s − 8.63·47-s + ⋯
L(s)  = 1  − 1.21·5-s + (0.357 + 0.933i)7-s + 0.140i·11-s − 0.842i·13-s − 0.218·17-s − 0.190i·19-s − 1.48i·23-s + 0.466·25-s + 0.784i·29-s − 0.234i·31-s + (−0.433 − 1.13i)35-s + 1.70·37-s − 0.860·41-s + 1.80·43-s − 1.26·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.933 - 0.357i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.933 - 0.357i)\)
\(L(1)\)  \(\approx\)  \(1.306315316\)
\(L(\frac12)\)  \(\approx\)  \(1.306315316\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.946 - 2.47i)T \)
good5 \( 1 + 2.70T + 5T^{2} \)
11 \( 1 - 0.464iT - 11T^{2} \)
13 \( 1 + 3.03iT - 13T^{2} \)
17 \( 1 + 0.900T + 17T^{2} \)
19 \( 1 + 0.831iT - 19T^{2} \)
23 \( 1 + 7.12iT - 23T^{2} \)
29 \( 1 - 4.22iT - 29T^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 5.51T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 8.63T + 47T^{2} \)
53 \( 1 - 8.45iT - 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 3.27iT - 61T^{2} \)
67 \( 1 - 3.40T + 67T^{2} \)
71 \( 1 + 7.76iT - 71T^{2} \)
73 \( 1 - 4.83iT - 73T^{2} \)
79 \( 1 + 5.04T + 79T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 - 5.22T + 89T^{2} \)
97 \( 1 - 12.3iT - 97T^{2} \)
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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

−8.580489352791660204106457614108, −8.087398696719829983752463408333, −7.44656172398063480157381113921, −6.51828120539836238608424133531, −5.68328110863071385705390353175, −4.80082024327217826939720676554, −4.14120219036959638543420764268, −3.08550967857014303975301134968, −2.30924621856386105049230904822, −0.73562319421716886607594677260, 0.65258723774628215767393936853, 1.89523129240823615708830330915, 3.29811531417900701803907666453, 4.03365242514426305878003587693, 4.50941010035164023812313520191, 5.57266496665455990280747238975, 6.63144472087559174307557039516, 7.29657031969394125579346577636, 7.892003757130957269764449692361, 8.444029561310224180735854490813

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