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Properties

Label 1-3024-3024.1859-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.925 + 0.377i$
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  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.642 − 0.766i)29-s + (−0.766 + 0.642i)31-s − i·37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.766 + 0.642i)47-s + (0.866 − 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.642 + 0.766i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.642 − 0.766i)29-s + (−0.766 + 0.642i)31-s − i·37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.766 + 0.642i)47-s + (0.866 − 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.191451877 + 0.4299894927i\)
\(L(\frac12)\) \(\approx\) \(2.191451877 + 0.4299894927i\)
\(L(1)\) \(\approx\) \(1.330633207 + 0.1490200497i\)
\(L(1)\) \(\approx\) \(1.330633207 + 0.1490200497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (0.642 - 0.766i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
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 \( 1 - T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.642 + 0.766i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−19.054053257524150972009565299589, −18.10324694109301922706491048974, −17.63290341891583654196680996637, −16.79216639834967175934262522483, −16.56207943716344551876017505750, −15.29871217101623660280951172263, −14.96661174110598798666561882535, −14.0733237441421073052174940407, −13.17693944388492898663499039120, −12.7733538095403067273083057985, −12.16798536170572198572149355982, −11.12062091404483854047468045840, −10.425251139737597662458446084645, −9.73449395586533079270708257925, −8.905991967803390698766039514345, −8.45103636655092515065111847307, −7.52139608876209198533642229463, −6.51254846608168672405426907730, −6.000870937139896663828028296437, −5.05770782010968664027561339643, −4.44648959518827850658655653601, −3.53616328588201049551650305217, −2.51097529803297510974450268683, −1.57835920862936302012163118866, −0.89551894632650335464035420749, 0.9144480348979751462865755332, 1.86633228427196862347450316127, 2.717380551632172527245265526548, 3.567105438932077829820251716794, 4.2479747560975030241303365512, 5.52758179766015551732443922523, 5.9393250825758791102876339541, 6.87357681482866580048930782605, 7.28854277248579027708837244801, 8.5704854751775016441027369139, 9.03383897557233546915272939664, 9.845234265441624329672003563754, 10.658654096613748521258344962068, 11.22641383359555504707859163701, 11.8961322504132156640273679947, 12.84952118931119973551946586268, 13.6979246174411955615949432420, 14.14646138204832341495990390830, 14.690003021981362138429911208853, 15.625643551001901799418720704066, 16.40652841076941715122302529216, 16.97795236851380124571172506060, 17.75040795779302302347548049212, 18.44180078213938628513731154743, 19.137689288577820036162966387959

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