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Properties

Label 1-3024-3024.1195-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.891 + 0.452i$
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  + (0.642 − 0.766i)5-s + (−0.642 − 0.766i)11-s + (−0.642 + 0.766i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.642 − 0.766i)29-s + (0.766 + 0.642i)31-s + i·37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.766 − 0.642i)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 + (−0.642 + 0.766i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.642 − 0.766i)29-s + (0.766 + 0.642i)31-s + i·37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.766 − 0.642i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.589216342 + 0.3801351707i\)
\(L(\frac12)\) \(\approx\) \(1.589216342 + 0.3801351707i\)
\(L(1)\) \(\approx\) \(1.123576195 + 0.02397156402i\)
\(L(1)\) \(\approx\) \(1.123576195 + 0.02397156402i\)

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 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−18.82058218764561362212149878594, −18.21234649094133366552814512498, −17.71658721818388462978970800924, −17.12576852229857042089905933186, −16.04931827363580102848658643533, −15.53237108195353889216496879721, −14.70498980961493904204969278382, −14.14515977583216794759087394542, −13.46805929991250471293669548814, −12.6355806389799007884514695873, −12.02657715641548849945852879626, −11.05890107535258561764776848726, −10.41498363729068083387120095278, −9.740046992630966873709673270152, −9.29665075993503348141017520773, −8.0048624640005650395576947714, −7.38112376905898554021668996208, −6.888105019849784602859567093832, −5.68267428106974333885012384293, −5.36379135595727889649206732062, −4.36011834277055357211059729767, −3.23977510916134958022479271796, −2.58760530319090340554990196305, −1.937264630415259728356439078513, −0.56182077746637113776682374919, 0.9463763721343937217286037358, 1.772829999965270149078839595837, 2.63610338980231432565924538792, 3.64051118588690700607354947815, 4.50394611259369507313195770354, 5.34649147171318434290779050222, 5.89084253782507366000600293149, 6.66243584515230876703336205666, 7.90853587543586860674281335100, 8.16958936672708792694830334132, 9.2031270166431831385688648439, 9.85271236597506217331323966788, 10.351551968370371321069494817784, 11.4969647939566870271577929335, 12.02565544620490890485320887814, 12.82798753337795713728664521350, 13.52661884509950362483428207957, 14.06190248405096697118816763233, 14.80555526865095545357982165248, 15.82694664782276530831053368685, 16.37883731724662044269400382406, 16.93689909963142650218027737548, 17.65402496783325493720562163542, 18.37996458598863948125266161047, 19.13469089261795973423475908269

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