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Properties

Label 1-3024-3024.1445-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.978 - 0.206i$
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

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.978 - 0.206i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.978 - 0.206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.941770508 - 0.2024506573i\)
\(L(\frac12)\) \(\approx\) \(1.941770508 - 0.2024506573i\)
\(L(1)\) \(\approx\) \(1.216454284 - 0.1200454755i\)
\(L(1)\) \(\approx\) \(1.216454284 - 0.1200454755i\)

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

−18.95273761378738918449869425016, −18.43096622963621011900261146435, −17.83920564331222505397122690753, −16.97167293249640323190945007622, −16.15111800389025489923776831253, −15.80466040183269651187155288044, −14.51560870154539452079025117379, −14.27766756958001610540845401108, −13.67236588724118324938236745283, −12.81488379195271504653028081931, −11.78908378095568371045125114, −11.24202505091506221753871692515, −10.715114035958350525811214380181, −9.73857914262908055077882807579, −9.1637814562211342586125428035, −8.376563473719840928304947298231, −7.373709959233952394105616953097, −6.74705004516917852323834401346, −6.111458417660754656276557619154, −5.33862478020970133574979085169, −4.2745489230454415326559191557, −3.48209708295502379316877439438, −2.726975540920755461004562885450, −1.91037201737232448536897478796, −0.776037825163297316113410383081, 0.88287474047739898927952731981, 1.59144153382738368138750464583, 2.55676903042100699784961496206, 3.58426312463108319379092940058, 4.47461195626452655368967728426, 5.07526467866979171936981201485, 5.921573727412822268780171611248, 6.595214927884979320241112157629, 7.761072029188265420010337171718, 8.11682632257965147841087217590, 9.24321304947940916246121911816, 9.559558804605875769509841004881, 10.41487139745146195338102173959, 11.3083150552298275544936401874, 12.07171952473947645055248863749, 12.76100526633741798689755477969, 13.31841575509671460426986130949, 13.94651894544733527819599766902, 15.036681737263394114758225751787, 15.51692134870839818809145005913, 16.16437433491937486350346568721, 17.220845567181259453988871900477, 17.50817780736410300712594615103, 18.040207728030299182154766288153, 19.24284127232296352430310613646

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