Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.0288 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.10·5-s + (2.64 − 0.0763i)7-s + 3.42i·11-s + 4.98i·13-s + 6.38·17-s + 4.65i·19-s − 8.98i·23-s − 3.77·25-s − 1.51i·29-s + 6.71i·31-s + (−2.92 + 0.0843i)35-s − 2.83·37-s + 10.1·41-s − 10.8·43-s − 12.8·47-s + ⋯
L(s)  = 1  − 0.494·5-s + (0.999 − 0.0288i)7-s + 1.03i·11-s + 1.38i·13-s + 1.54·17-s + 1.06i·19-s − 1.87i·23-s − 0.755·25-s − 0.280i·29-s + 1.20i·31-s + (−0.494 + 0.0142i)35-s − 0.465·37-s + 1.57·41-s − 1.66·43-s − 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0288 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.0288 - 0.999i)\)
\(L(1)\)  \(\approx\)  \(1.614846047\)
\(L(\frac12)\)  \(\approx\)  \(1.614846047\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.64 + 0.0763i)T \)
good5 \( 1 + 1.10T + 5T^{2} \)
11 \( 1 - 3.42iT - 11T^{2} \)
13 \( 1 - 4.98iT - 13T^{2} \)
17 \( 1 - 6.38T + 17T^{2} \)
19 \( 1 - 4.65iT - 19T^{2} \)
23 \( 1 + 8.98iT - 23T^{2} \)
29 \( 1 + 1.51iT - 29T^{2} \)
31 \( 1 - 6.71iT - 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 3.02iT - 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 - 9.16iT - 61T^{2} \)
67 \( 1 - 5.07T + 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 - 7.79iT - 73T^{2} \)
79 \( 1 - 2.05T + 79T^{2} \)
83 \( 1 + 5.73T + 83T^{2} \)
89 \( 1 + 6.47T + 89T^{2} \)
97 \( 1 + 8.04iT - 97T^{2} \)
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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

−8.685962196651197681937809611444, −8.184102732800109278039761935546, −7.46509813771615139221933735060, −6.80005688291895389290395176572, −5.86274987971103483537224239973, −4.82774363470226042598157395903, −4.37057814806910121060507028164, −3.46583551380212325973682174755, −2.13449372053034542620694500917, −1.36231360291101033906958035991, 0.53463040668040283487237287676, 1.64129680208098827805156591145, 3.11702148687753301236754344814, 3.53970822476060930937175042580, 4.78284358445740780027154708614, 5.47495759299662818384845203460, 6.02148172044000383887353669218, 7.35549269336576362094730382705, 7.930717435902921698677118462760, 8.179447317799400977545397800104

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