Properties

Label 1-3024-3024.403-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.968 + 0.249i$
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  + (0.642 − 0.766i)5-s + (0.642 + 0.766i)11-s + (−0.342 − 0.939i)13-s + 17-s i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)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 + (−0.342 − 0.939i)13-s + 17-s i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08263029953 - 0.6511400698i\)
\(L(\frac12)\) \(\approx\) \(-0.08263029953 - 0.6511400698i\)
\(L(1)\) \(\approx\) \(1.051689931 - 0.2411147717i\)
\(L(1)\) \(\approx\) \(1.051689931 - 0.2411147717i\)

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 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 - T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−19.09236076056855719789562712595, −18.61330021873340742204635862685, −17.98119057061078540954949836357, −17.02151619953717982704752166659, −16.60719192462601311085914027434, −15.89006383555908574794215310708, −14.74817925667839075513151565840, −14.19436998532860229887048619881, −14.051803152337533278030614811483, −12.92416177737345069654573036708, −12.1118476248258065377010160417, −11.484569768345927551710862569844, −10.695754715347020522012266110946, −9.965456101620041235786199879024, −9.41139097122836334220688545124, −8.54704317681158759917901097855, −7.66758300688853060588681648530, −6.916282352648636631894621210845, −6.074761863281273328065613346258, −5.731412429852222930348037512691, −4.53576161930996336717981794595, −3.647248026508330419113285613987, −2.99949167930172721343026171036, −1.94053245483018399335439046261, −1.3148327527371411680305437404, 0.09756395942199581221514474104, 1.09860642519869917518996272366, 1.84632843592784506524963377894, 2.78508078065413581109417288399, 3.74726608513607334229226651582, 4.72990093881826633954747089333, 5.24616730412129446620242067668, 6.068616690725506412224539387211, 6.8743344081935668426729941254, 7.78240150168047344229067451411, 8.438291871447144628450551301739, 9.35008344346133591894667416793, 9.85999224411825666645010050821, 10.46080822933838807280107253947, 11.58148876676673529853326111997, 12.33848943317175829050544646444, 12.6737047038878971504837741231, 13.6891117862999786191936020999, 14.1193626449543322366540270751, 15.08835303285883007601038652420, 15.669576446472598921785735576547, 16.50667268616168150812392221225, 17.304576514488433447349587785392, 17.57481543900134700393763938243, 18.329921963597721573761441017059

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