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Properties

Label 1-3024-3024.1213-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.796 - 0.604i$
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 + (−0.342 − 0.939i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (0.342 − 0.939i)29-s + (−0.939 + 0.342i)31-s − i·37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.939 − 0.342i)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 + (−0.342 − 0.939i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (0.342 − 0.939i)29-s + (−0.939 + 0.342i)31-s − i·37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.796 - 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184866561 - 0.3987769976i\)
\(L(\frac12)\) \(\approx\) \(1.184866561 - 0.3987769976i\)
\(L(1)\) \(\approx\) \(0.9880584181 + 0.05702099403i\)
\(L(1)\) \(\approx\) \(0.9880584181 + 0.05702099403i\)

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 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
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 \( 1 - T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−19.23136324733709197747694434661, −18.42937980214772672430566334382, −17.67691562714712814116825967093, −16.8173049374355343186827095807, −16.48102106723208400823229255758, −15.87443448282366736820212584488, −14.72449363341407921416180722872, −14.31765781486578732044496237471, −13.316675806568239751437439531305, −12.879484923057616198054472890963, −12.18929240882320167140628189120, −11.322986619373523808225430931, −10.594011631179747446381431386912, −9.86130016804794420338379102878, −8.8546942923151504151913406402, −8.60892740741000897493414756958, −7.778764188068008184204624652685, −6.63109768967811034581221901649, −6.07242211713473944181779234211, −5.2619697211957496540560930274, −4.40872540322446481013431283961, −3.8454578443604296850075083110, −2.58148759115334920627040075485, −1.85577741595268862470168481875, −0.89808311479884953172713454855, 0.43253649722067137559991100856, 2.03473734295902877592661921979, 2.41413174739883470018423339557, 3.35069279444642189529456463569, 4.26427428067344351586335266750, 5.192758305393797420483585443099, 5.85474310560983043879542328141, 6.83145238842389523345724365962, 7.34360027808891653803503342473, 8.010312728174951432290295909800, 9.19913322372488764525552210235, 9.71503830402242275512942559256, 10.514262876220155795686471168829, 11.03466007271096597154674594312, 11.86212737966181394599159736100, 12.79224676468601094223688086247, 13.275232058946128979199857691225, 14.193270406541239264634723284879, 14.76506103556012234088183431045, 15.511258871413311550788319773394, 15.90127881548873177944751482546, 17.22112702169478296691444572649, 17.67751757510365773624703813279, 18.07394519993462864983597193990, 18.96866253120617706071597630095

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