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Properties

Label 1-3024-3024.11-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.00934 - 0.999i$
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 + (−0.984 − 0.173i)13-s − 17-s + i·19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.984 + 0.173i)29-s + (−0.766 − 0.642i)31-s + (0.866 − 0.5i)37-s + (0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (0.766 − 0.642i)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 + (−0.984 − 0.173i)13-s − 17-s + i·19-s + (−0.173 + 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.984 + 0.173i)29-s + (−0.766 − 0.642i)31-s + (0.866 − 0.5i)37-s + (0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (0.766 − 0.642i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00934 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.00934 - 0.999i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.00934 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5574283213 - 0.5522423204i\)
\(L(\frac12)\) \(\approx\) \(0.5574283213 - 0.5522423204i\)
\(L(1)\) \(\approx\) \(0.9105936056 + 0.03685623330i\)
\(L(1)\) \(\approx\) \(0.9105936056 + 0.03685623330i\)

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

−19.44409276535592364756855848373, −18.385747386210301842122125665923, −17.65768454497650948990127492223, −17.22639973879946797729621434333, −16.48526262591515948797787856571, −15.841451031535013934400478845734, −14.87495946518117168586014497493, −14.51783898566446916133706068300, −13.33479521656652031216240731160, −12.976687832310922207555499734211, −12.250490461238714651431499074584, −11.555506192553983089367108280447, −10.709030102408983029543829717342, −9.66653617712971246890893500246, −9.38277048229675968859838151808, −8.59141666874334640349434043026, −7.722847195582134680742213720158, −6.897855800461514932990836682253, −6.252938422233160247847552485667, −5.10532744132841753082232049832, −4.68013815282939828934309631530, −4.02078486586689391215401899537, −2.60744941902916596888694186780, −2.06070652470645717653233008180, −1.05009399648599357050375750994, 0.2399102672298332877515366345, 1.789761024647418761115972597182, 2.34037173143087781543970903801, 3.44411143075195775631452149007, 3.87069568249514866737122434077, 5.16308650365237396654403817947, 5.84655877140483721165523511958, 6.492495434173650286308109101830, 7.40321479207451829297114660094, 7.87349659486441712028831395950, 9.09597305031455962574984436153, 9.504991067847357864943409980397, 10.49302557393954627257261435111, 10.98891285889188535130687469234, 11.70737693373061021186889119952, 12.5025855766773272347268851562, 13.49231844715760661412095101565, 13.88957395918497767895775300241, 14.78822814765541363203102772522, 15.14723095756470732483304622707, 16.11656169552417213705424010974, 16.92135107785317460665520070744, 17.42631260105182054786253564656, 18.319283691893996319713217606269, 18.77177864430256513573891886089

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