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Properties

Label 1-3024-3024.1811-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.206 - 0.978i$
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  + (0.342 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1905294011 - 0.1545510206i\)
\(L(\frac12)\) \(\approx\) \(0.1905294011 - 0.1545510206i\)
\(L(1)\) \(\approx\) \(0.8680310773 + 0.2783046260i\)
\(L(1)\) \(\approx\) \(0.8680310773 + 0.2783046260i\)

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

−18.86733505453209104011527635245, −18.475996568859258268991833001716, −17.40750385373783042410999706718, −17.00054472346443722578226723041, −16.32099729175561798974395637203, −15.57595088728057317282402390210, −14.92247703494848357912764297339, −13.99544285565292268042343340541, −13.23007423778525898183067428180, −12.83716013035202504464947689226, −12.14873302126300417885917215405, −11.06827099396082643829140641656, −10.58470860094723628339556302800, −9.75877383743484568543958530193, −8.87368356768139138107235313809, −8.24533159906684592539376006373, −7.87256110649528694024094684229, −6.3921663641106814590928916754, −5.993638620187686436558295544151, −5.22612210631781890009047479792, −4.33043858772330541308193985133, −3.61825391758673827135341172649, −2.54419343771384938935938169415, −1.70391217421568116461852928328, −0.71269561195915935518731907090, 0.046321964086972761382264826976, 1.63350499962149438436023880817, 2.174804255980751830161776199326, 3.06104503334773725777885537521, 3.9737823691083363520713659527, 4.77097263204167986611858230524, 5.663367793524943420034831049226, 6.520204077603901174225313013093, 7.17700733895444103397595131734, 7.61545957561946790558732206367, 9.059988589952366886253556915401, 9.25665347087880725355019366078, 10.28754281433658436566587521707, 10.929037129515315582714378036203, 11.47628092888609887553579602229, 12.407347233887490312378984107, 13.17980281548319507339008799493, 13.8814733451130266237537440012, 14.50376748221260941227711244078, 15.20000175328069715801682520745, 15.83410425252453403373215440967, 16.64085117677354638964709664609, 17.53484695264646763398494673139, 18.08076393600447267870223144042, 18.53056812410156550066074759074

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