Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.888 - 0.458i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)5-s + (2 − 1.73i)7-s + (1.5 + 2.59i)11-s + (−2.5 − 4.33i)13-s + (1.5 − 2.59i)17-s + (2.5 + 4.33i)19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (−1.5 + 2.59i)29-s + 4·31-s + (1.5 + 7.79i)35-s + (3.5 + 6.06i)37-s + (−4.5 − 7.79i)41-s + (5.5 − 9.52i)43-s + (1.00 − 6.92i)49-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s + (0.452 + 0.783i)11-s + (−0.693 − 1.20i)13-s + (0.363 − 0.630i)17-s + (0.573 + 0.993i)19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (−0.278 + 0.482i)29-s + 0.718·31-s + (0.253 + 1.31i)35-s + (0.575 + 0.996i)37-s + (−0.702 − 1.21i)41-s + (0.838 − 1.45i)43-s + (0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.888 - 0.458i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (2881, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.888 - 0.458i)\)
\(L(1)\)  \(\approx\)  \(1.785916447\)
\(L(\frac12)\)  \(\approx\)  \(1.785916447\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.560354481034445039471003182280, −7.84063556711123871832434234637, −7.22049715118263417289468008044, −6.92477729801594174902433456730, −5.65101826124369984931919018617, −4.88645056213009144906849478884, −3.95255198246004845921944366070, −3.22912804504060757296101278912, −2.27675496325682002985285286244, −0.872358056979913534653660343269, 0.806487909841070155741251105632, 1.81911240424870450360908710073, 3.02740724669832917063151796102, 4.22249335869452173749016267614, 4.67639345506329846383875361261, 5.48079937631410340763345740791, 6.31173873539867523202513171634, 7.35960431667286084510556057920, 8.042413110337622028823033812340, 8.691703799719210854682480510376

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