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Properties

Label 1-3024-3024.1237-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.998 + 0.0622i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.998 + 0.0622i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.998 + 0.0622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4568729719 + 0.01424235649i\)
\(L(\frac12)\) \(\approx\) \(0.4568729719 + 0.01424235649i\)
\(L(1)\) \(\approx\) \(0.6790877003 + 0.1017054985i\)
\(L(1)\) \(\approx\) \(0.6790877003 + 0.1017054985i\)

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

−19.03852609040104003735861138962, −17.97401020753251649358861994212, −17.398047336042433949617699026527, −16.95018602958580077080670040861, −15.95164775402523596610355646498, −15.38301211674765785319190302263, −14.93708134958665898417436798798, −13.79888592665043842003313857552, −13.1055445978186321290744238204, −12.60933258389443034230666303027, −11.85845063406553323545914487790, −11.19091694261800996821400886832, −10.28569184963092274510302931199, −9.31288915823877068762639123944, −9.1254762241984373337413695894, −7.960643674171136226039834076583, −7.42576756662777005566151332605, −6.71946103139414725100451901495, −5.475455217867790530001472352207, −4.967236896692384684665604898566, −4.31476756414663526571820430657, −3.40351436150865486199902196608, −2.23723939372162307663946052947, −1.656838343433043022810386168283, −0.28997629229607427945011263878, 0.199755442035074805703223551621, 1.71401194543999437490963648331, 2.52957655120036012980993601966, 3.2981463856816411608847315746, 4.04824762583092224952809477825, 4.98173156117442622345265170219, 5.89188412697367517478783759806, 6.61887034526812828521872549373, 7.29300712404033826120031058530, 8.08161736112772553924261328530, 8.74333986454831673029806811204, 9.774133531660436631457015878400, 10.399581402580081738240108194475, 11.11939666381186045341190965943, 11.64647869080925326274548165010, 12.550801948009881276040428392201, 13.257886044437184664143039267201, 14.18418535161006588402239689083, 14.63310780183182865591460720461, 15.254526193361130602245639233390, 16.274671057308385563504086485841, 16.53793632144047311190423892116, 17.70096515543627578378629784479, 18.190529261127615774946677959801, 18.95995087191549281318341570109

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