Properties

Label 1-3024-3024.1139-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.661 - 0.749i$
Analytic cond. $324.973$
Root an. cond. $324.973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (0.984 − 0.173i)13-s + 17-s i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006833092410 + 0.01514381469i\)
\(L(\frac12)\) \(\approx\) \(0.006833092410 + 0.01514381469i\)
\(L(1)\) \(\approx\) \(0.9369636265 + 0.2083542226i\)
\(L(1)\) \(\approx\) \(0.9369636265 + 0.2083542226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (0.342 + 0.939i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.21561532446567525823877579484, −18.50679325304591860037455603855, −17.68593686535595845790711496882, −16.86295044811338914081258462328, −16.38978840670682790764827950941, −15.76204452381900012168053187577, −15.071835766708965583549308778608, −14.041491428591249395073560132309, −13.499609925683843288525822648438, −12.87032050794561716981160209831, −11.99959984502640158359007514365, −11.37821632913475702889574865343, −10.836869830359524816459311178416, −9.59302555005258533152865918259, −9.17901338909712077130696852697, −8.30474585985372193481798047186, −7.864824353516494084527613296259, −6.81114477308474813745636983421, −5.922018854709914261573356084833, −5.33535802434748972518163400152, −4.43040329523358872755810906327, −3.62609940591382688461454547037, −2.99304233766077213546766626227, −1.51925086737132611925442864744, −1.04926227254675265547693430384, 0.00278367813663501076376971806, 1.26631917670373527495593294234, 2.125180783651738078057258462003, 3.11960801913886901575277476565, 3.79110739743970373698999864941, 4.49796133655463449833581484440, 5.66931866713769742031772473251, 6.3203947924008722393014501297, 7.007101955364407455237620659605, 7.90697601660303521737941169345, 8.301794408683782488134066996636, 9.599223361652210190165952332367, 10.03558043930233735336843069686, 10.80361542425302639231720535588, 11.55558070192253348324495349861, 12.17516013134203320967153911092, 12.951134706030861648286327836361, 13.81928746115333578462857789145, 14.64071379531563332888856415278, 14.906854705483733815613053143712, 15.79225124103383710523495164005, 16.51368643132243216952697419192, 17.233375234316548114806170075343, 18.135001900778899125373765580403, 18.623187079932611456415039691538

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