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Properties

Label 1-3024-3024.2797-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.713 + 0.700i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.713 + 0.700i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5917332992 + 1.447669663i\)
\(L(\frac12)\) \(\approx\) \(0.5917332992 + 1.447669663i\)
\(L(1)\) \(\approx\) \(1.025996656 + 0.4495695165i\)
\(L(1)\) \(\approx\) \(1.025996656 + 0.4495695165i\)

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

−18.758588410199228779057628604911, −18.173967689257034772198351401320, −17.14772630068044658218251482306, −16.8544877199778230053960331747, −15.79079646948228597353646437538, −15.73165369653656916235701091462, −14.36927255286347918275626394008, −13.85462483114262825008166799088, −13.068732254421988693355886546699, −12.66370112551344502465173141465, −11.66095594575347705210171606006, −11.044971688069843771036487352183, −10.21655798986858839042139552432, −9.4490592529336465910998710118, −8.60518710600446952982172249734, −8.28195615199078467279313454446, −7.31313531543214144661468533451, −6.20504498256958836380408259942, −5.710573121925204271312052257306, −4.96841380419217864446168164883, −4.073081111306442199676663021076, −3.24682963575994566954487739875, −2.30315023070268410320878683517, −1.24570970992751387740085061408, −0.48366465676829916925234914257, 1.45513596717226481984272207555, 1.94501958526978466580432957018, 3.228045960266460795952790695, 3.54764430201065139022335555435, 4.70920315748292512479262290551, 5.65188530973360067648162361099, 6.191137463266872738393547672989, 7.1305753169708716104733561859, 7.65760996739856073527447206745, 8.51467273946122289831265863075, 9.54668921645486328356502214130, 10.11376765209953909046093610131, 10.65332463249789672971549605196, 11.55942706172517704826626346749, 12.1815745650687231326331690900, 13.05559888344104224542498811657, 13.86047019505759055946648404664, 14.28969956703683547243427781105, 15.26861697923211807532637871428, 15.544615128879594232607250492689, 16.66065876990556052096199462141, 17.21010369416589222665682675156, 18.20371883995849340021018934663, 18.420405238843505921628013731736, 19.14756158862427886806596618384

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