Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.391 + 0.920i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.86·5-s + (−2.43 + 1.03i)7-s − 3.32i·11-s + 0.821i·13-s + 3.52·17-s − 5.25i·19-s − 5.12i·23-s + 3.22·25-s − 1.64i·29-s + 2.55i·31-s + (−6.98 + 2.96i)35-s − 8.93·37-s + 3.49·41-s + 0.161·43-s − 5.34·47-s + ⋯
L(s)  = 1  + 1.28·5-s + (−0.920 + 0.391i)7-s − 1.00i·11-s + 0.227i·13-s + 0.853·17-s − 1.20i·19-s − 1.06i·23-s + 0.645·25-s − 0.305i·29-s + 0.458i·31-s + (−1.18 + 0.501i)35-s − 1.46·37-s + 0.546·41-s + 0.0245·43-s − 0.779·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.391 + 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.391 + 0.920i)\)
\(L(1)\)  \(\approx\)  \(1.876322005\)
\(L(\frac12)\)  \(\approx\)  \(1.876322005\)
\(L(\frac{3}{2})\)   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 \( 1 + (2.43 - 1.03i)T \)
good5 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
13 \( 1 - 0.821iT - 13T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 5.25iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 + 1.64iT - 29T^{2} \)
31 \( 1 - 2.55iT - 31T^{2} \)
37 \( 1 + 8.93T + 37T^{2} \)
41 \( 1 - 3.49T + 41T^{2} \)
43 \( 1 - 0.161T + 43T^{2} \)
47 \( 1 + 5.34T + 47T^{2} \)
53 \( 1 + 3.28iT - 53T^{2} \)
59 \( 1 - 4.08T + 59T^{2} \)
61 \( 1 + 8.43iT - 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 + 7.86iT - 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−8.858375953836607988861986690153, −7.932788778893536693675801037316, −6.67247467996789365429934992211, −6.44081065342764636528694092986, −5.55626751255773744378451639879, −4.98273666155388259885197590450, −3.60630497749808257096703724150, −2.85385110892925448808616867966, −1.99649355977867010398874918688, −0.59307647633699720773697149638, 1.28458130888012680358315141416, 2.16967813702543475395217811564, 3.26252767111316389268736316248, 4.04196159390263276846896198324, 5.31293915851049103333497344069, 5.72739021183795225546179776555, 6.58612711953300870242128751404, 7.26305788620555275232202076712, 8.037030456397459749301036292684, 9.117673759182142040339486384254

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