Properties

Label 1-3024-3024.2531-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  + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.642 + 0.766i)13-s − 17-s i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (0.939 − 0.342i)31-s + (−0.866 + 0.5i)37-s + (0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.939 − 0.342i)47-s + (0.866 − 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (−0.642 + 0.766i)13-s − 17-s i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.642 − 0.766i)29-s + (0.939 − 0.342i)31-s + (−0.866 + 0.5i)37-s + (0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.939 − 0.342i)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: $\chi_{3024} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774937468 - 0.7255036357i\)
\(L(\frac12)\) \(\approx\) \(1.774937468 - 0.7255036357i\)
\(L(1)\) \(\approx\) \(1.244872897 - 0.09791457154i\)
\(L(1)\) \(\approx\) \(1.244872897 - 0.09791457154i\)

Euler product

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

−19.325088424452421906257993849753, −18.1458106919660119985972802503, −17.74141340072945665804520519844, −17.179337681046261890779504594028, −16.44629547992417149161738410629, −15.68878577762736882947171810730, −14.739894591178751319095305794509, −14.28802843453208828722604116529, −13.55070974765115825668739294358, −12.77717873023980506578564409481, −12.21138113607166664753599803774, −11.37344494951885644947759986717, −10.41519535683456020928787872133, −9.92466922066259225784224224220, −9.161194747776006939596249980713, −8.55260630422829881621584441270, −7.546821319137491202353046160593, −6.7878139975727194872468735158, −5.98278122194822413679567126795, −5.420498303008932261690358474855, −4.48658347870655835686271429934, −3.68113627084402787114196931964, −2.63182588400539294381677183165, −1.86190154649943640081442771444, −1.06344145050192337809110804006, 0.6057064522379137649652225837, 1.96507960412778676176554952427, 2.24128119071903572808867014158, 3.40395497831000090180923340284, 4.40848021479381277998270114158, 4.96354068820108539282136676330, 6.12931786339973306356069480079, 6.531010394051502403929346599345, 7.20030615557259758678958121386, 8.369550200692310257255354423400, 9.10414107390846790500969154836, 9.60083905293496700107742579652, 10.34117080706820899696223426249, 11.26064721598029812449603801219, 11.78927138931090437964441245152, 12.68254278311848232786940316175, 13.575611757418021411411139104563, 13.89873454969811774151333760634, 14.73041743437952686909736687975, 15.33552456590614371378951761994, 16.35647267100639999785370082570, 16.968440145954076474442987305356, 17.56363225347111826759272600127, 18.08831268365821809900997005790, 19.10471442866183089727011700018

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