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Properties

Label 1-3024-3024.1955-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.921 + 0.387i$
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  + (0.984 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.579568655 + 0.5200806993i\)
\(L(\frac12)\) \(\approx\) \(2.579568655 + 0.5200806993i\)
\(L(1)\) \(\approx\) \(1.498245156 + 0.1258792123i\)
\(L(1)\) \(\approx\) \(1.498245156 + 0.1258792123i\)

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

−18.734630866483097577080088068362, −18.37023030383706209748198479206, −17.49521998352252456552499864535, −17.11427740217127372642475294787, −16.14818649244920899214713887671, −15.703127731163753697357869345882, −14.62550291060319491805132324121, −13.96266574285058144968972658320, −13.56368002651786123010302555947, −12.76611405366358022212445863108, −11.79835361438467188284081587755, −11.399428956332425432502956422225, −10.290284946006970112534481047181, −9.69464878888822461957047585127, −9.136793223446412459567359758341, −8.35608694299391638732685239526, −7.40961803378113607150940953388, −6.57578706926068489926054411051, −5.94239500054222215127489104153, −5.23010459923520327929702813037, −4.36352045380811877069896591218, −3.408936908154114343258070102639, −2.621567187870236273895760112865, −1.53891362425216540616354096728, −0.95820493775130339306345932749, 1.15508084974970068763522568782, 1.61845607937532105697864494851, 2.72882997678520640878392420145, 3.595462685450089782586156347685, 4.31370361974572705505109877320, 5.450458764894709146894360797394, 6.105106553899287056157501683311, 6.52862072078783825823800908538, 7.56517334818187639466549260023, 8.49429688183489346253540195941, 9.089890568071293652294364331449, 9.85069905534835497736499618619, 10.54921897278608350737525448229, 11.17169895017343060924688709991, 12.21171013707098933318793274941, 12.64188345833119720999109009476, 13.72717125924461696685172110443, 14.17004169238474073543090132714, 14.58002245805231450408160531806, 15.780511252160627158323547547071, 16.313029743827226445412456856650, 17.057507559228845361191708487346, 17.77440628774661869037028680145, 18.24569944863717760003847588595, 19.08819863323788398320778663218

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