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Properties

Label 1-3024-3024.1427-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.561 + 0.827i$
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  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.173 + 0.984i)47-s − i·53-s − 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (0.342 + 0.939i)11-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.173 + 0.984i)47-s − i·53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9675677172 + 0.5126611481i\)
\(L(\frac12)\) \(\approx\) \(0.9675677172 + 0.5126611481i\)
\(L(1)\) \(\approx\) \(0.8353227687 + 0.1083492185i\)
\(L(1)\) \(\approx\) \(0.8353227687 + 0.1083492185i\)

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 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - iT \)
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 \( 1 + (0.642 - 0.766i)T \)
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

−19.10986614017255794733212424053, −17.97854407863868750868053020749, −17.176248338105446604820068928805, −16.69044735004650783459140682785, −16.14949099260222256616633914392, −15.26092776429483642514427942595, −14.651071826793192181579835655864, −13.69756973781395413674209949126, −13.21293162273261274084143135071, −12.29212891768841927985907670155, −11.83443436309118819881513075716, −11.0591848750800360048284759507, −10.21760289752260979126827210391, −9.32925345193971087550860255431, −8.66728359968262127523426434134, −8.20068696926786138786672314483, −7.240780533787953795422388943669, −6.37538058530497626040380196394, −5.637980565252194755124129868729, −4.789397293000690664263829868508, −4.02692886866107400984612972470, −3.425616223288369251878855509367, −2.08192712375116963215899285776, −1.44095219620894390584863588633, −0.31496643625495581891763737937, 0.473656926202133115264256648912, 1.88624326786296157503403934622, 2.64640422300074680277527569120, 3.28101861393586533581277449587, 4.48936118474071054415461769396, 4.80554912110037122075899000638, 6.117728655063786215483605738305, 6.794193240317756912074344828238, 7.28568632800776058878006415893, 8.15703053381970639557122799507, 8.93799166347683484520449126194, 9.9793395925095069094975602362, 10.3458045519897829610828036616, 11.158812470923445713641621534611, 12.00058971810872899442648742318, 12.461295133682597682745001756988, 13.435516629260262059802229027, 14.21212409410117886818519454806, 14.82794369087819468639762144248, 15.44800047071926312332278261361, 15.94219429888639433547419992699, 17.24167354398454525422912515291, 17.456961269208368907329487108566, 18.35245154181582038444070851764, 18.89809259894848951365150945583

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