Properties

Label 1-3024-3024.635-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.445 + 0.895i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.237920768 + 1.386787602i\)
\(L(\frac12)\) \(\approx\) \(2.237920768 + 1.386787602i\)
\(L(1)\) \(\approx\) \(1.225246623 + 0.2229405475i\)
\(L(1)\) \(\approx\) \(1.225246623 + 0.2229405475i\)

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

−18.63802982035615423453248391128, −18.27027713694113619040908150064, −17.09011157823907049739661719475, −16.66068681173114250622512187217, −16.242109260907344464973120433819, −15.30112857744290787303946676616, −14.32423481173522647414468275352, −13.9003595551358993586654393211, −13.07531381351691089999923830086, −12.5472789477020733084833945116, −11.70502360127458845694208013335, −10.9504065476709657934515875983, −10.0856017018286353132477032621, −9.53139908235312856687140556030, −8.58109832603241610385870302120, −8.250265526936211529119151841421, −7.15927324448780043708385898999, −6.30618004662489356946221539271, −5.51348794811992302986095333382, −5.06394218568530271187364033126, −3.98097115562460538861712352316, −3.224362527288642543705135798707, −2.14042096513908824552215835301, −1.372706877071513127742405510709, −0.51060383408546252802199072687, 0.7799184831680491982718127294, 1.76175555911759449559801704035, 2.78710891243602503977938345172, 3.14759753158771677859120136915, 4.34400609191867953349917878784, 5.36752540462612974884659969620, 5.71981104642263177984178118352, 6.86154053643920025919042899255, 7.31922495073978643323916725766, 8.142962092352390215986357233876, 9.11229846535643974481320314506, 9.876062835012705577989840278627, 10.43302189122641473985051677041, 11.03744725124317117186303761975, 11.916847476286841376553607556869, 12.890441285316079832671300353674, 13.26625669957952465892670333415, 14.12373455719031256212516972258, 14.8740126813335150711311824382, 15.386531881405292050780182401998, 16.08314275372702149350977087358, 17.21351516040989812134028942429, 17.60805232068833890460572801850, 18.20748183252897118578083820782, 18.93860708157127437834997514742

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