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Properties

Label 1-3024-3024.1915-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.508 + 0.861i$
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  + (−0.642 + 0.766i)5-s + (−0.642 − 0.766i)11-s + (0.342 + 0.939i)13-s + 17-s + i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + (−0.173 + 0.984i)31-s + (0.866 − 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.173 − 0.984i)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 + (0.342 + 0.939i)13-s + 17-s + i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + (−0.173 + 0.984i)31-s + (0.866 − 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (−0.173 − 0.984i)47-s + (0.866 − 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.508 + 0.861i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1915, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.508 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7660100212 + 1.341204802i\)
\(L(\frac12)\) \(\approx\) \(0.7660100212 + 1.341204802i\)
\(L(1)\) \(\approx\) \(0.9103952148 + 0.2463036583i\)
\(L(1)\) \(\approx\) \(0.9103952148 + 0.2463036583i\)

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

−18.621452014937188853808686550318, −17.92269600585655893944228877540, −17.259433078517670050648438718930, −16.485118413571014792608037662225, −15.78252940538428635115852796814, −15.30058501361381361573546044951, −14.594554621619353793832124275683, −13.53540191406839015141579591433, −12.88306015294152092809275914578, −12.44628394752749744787027385501, −11.58015798894958177174605181461, −10.93639511205810901477447513002, −9.95209390238384627645922205530, −9.46621807093019364965979478937, −8.434769430411492440246771612983, −7.77739321274029861714241114874, −7.443100193976845253771320066532, −6.10819534333116028009897675643, −5.50484375468805253943630711792, −4.60190728842006524282217842761, −4.06229731462890250359840266168, −3.01421474576323627955203278672, −2.20361379716239684241464447660, −0.97676304397185063285245203899, −0.34888047860172782248747492978, 0.81564490710329978991853820079, 1.90145803086738363953332581510, 2.86210598724448556689776722984, 3.66427733153889103807668848769, 4.14131110741332867623144699899, 5.463894341714901036024080623334, 5.94480873393790402158573853654, 6.94484639087606294893959610491, 7.57492701188736062224045960411, 8.26066492503604931419070354214, 9.00480834338648283191465874098, 10.081763254143702702424613219271, 10.55968840129077451507456319781, 11.36833670491705814697623322336, 11.93053439440203756642127731512, 12.68018786062696426215772264441, 13.639799408739895307374557207564, 14.42953329760020330731468784928, 14.608089409407095051707947488923, 15.94282942455286830168451878914, 16.09455492627015678742553682544, 16.82005571726811522739857301050, 18.11421164090278746669877768279, 18.314784846624347850254273865308, 19.15235714441760123633225334919

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