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Properties

Label 1-3024-3024.1307-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.452 + 0.891i$
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  + (0.984 + 0.173i)5-s + (−0.984 + 0.173i)11-s + (0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (0.173 − 0.984i)31-s + i·37-s + (−0.173 + 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)5-s + (−0.984 + 0.173i)11-s + (0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (0.173 − 0.984i)31-s + i·37-s + (−0.173 + 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7765761992 + 1.264887378i\)
\(L(\frac12)\) \(\approx\) \(0.7765761992 + 1.264887378i\)
\(L(1)\) \(\approx\) \(1.129541258 + 0.1123900776i\)
\(L(1)\) \(\approx\) \(1.129541258 + 0.1123900776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.984 + 0.173i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
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 \( 1 + T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−18.36822112910426212764886323084, −17.99975397116902456362570625449, −17.425474085430759619231341990354, −16.52248666184707496970363607151, −15.83847639847235032248171264446, −15.320711245132698295041283376284, −14.22587320022150436604087739818, −13.70284885239614293617506093761, −13.062475918426069649953868168299, −12.56344447493096781663254763470, −11.39509443085238432653629584436, −10.79670011439694600187226128673, −10.13102291299988904257362808566, −9.36645084516015936337788990093, −8.653880254012051748596274238673, −7.940981093522291358846419477489, −7.041577046312548664738573190423, −6.099672117735247649480616177171, −5.61335173343343741768455796557, −4.90003752536310594520722211274, −3.80836168441711072512522277403, −3.0139031455573780703915647657, −2.04447181563602717117703253925, −1.378605014596585627240247374869, −0.22972987750108974639955754511, 0.970065664543192854914887357504, 1.92758800634633745208376515427, 2.62740161909968644016713045847, 3.487272341124899177785004718063, 4.532941188438935952281195486301, 5.37520032072133500737613047125, 5.94645698966561924802122567110, 6.74439882952464779918356631089, 7.55611589904857479096989974409, 8.39515666942477227703730501587, 9.16218450966386341591486218182, 9.95571551317592516958997202941, 10.41019047281869569881784187247, 11.31135367224207001728255977922, 11.96312570980554924444186492529, 13.06342797724108513077469032718, 13.4795537835510166432743147275, 13.98383651960498375587960151145, 14.95135476335795360343709035623, 15.58320832824168433126340333179, 16.4601987277446791280539585828, 16.86648397502790249784704334831, 18.06062207871759096877434317397, 18.3148467128700669896233974598, 18.6807370615401223634259876006

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