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Properties

Label 1-3024-3024.1021-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  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (0.939 + 0.342i)31-s + (−0.866 + 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.939 − 0.342i)47-s − i·53-s + 55-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (0.939 + 0.342i)31-s + (−0.866 + 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.939 − 0.342i)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: $\chi_{3024} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.445461812 + 1.825561536i\)
\(L(\frac12)\) \(\approx\) \(3.445461812 + 1.825561536i\)
\(L(1)\) \(\approx\) \(1.505745969 + 0.3044359620i\)
\(L(1)\) \(\approx\) \(1.505745969 + 0.3044359620i\)

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 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
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

−18.726341254334273140711070890806, −17.91978674371569223958701516856, −17.40294950949412990722186719617, −16.76683364376130187691412835152, −15.89503663478515787040157800804, −15.53506147892279024967810324324, −14.27180103051734923980933059948, −13.931552566158868232048112395817, −13.19226306207670625507375069438, −12.350652878257990837171232199867, −11.87987018434833663460233429665, −10.95430952784360345563667637690, −10.11923063682549235597842343013, −9.286302946712484994854495492446, −9.01756078972757141740499770716, −8.0421409092217426269710965318, −7.1353815531492621880649951820, −6.43037142678987668008447283516, −5.59414127347607014205671167344, −4.84351295453140487955052137473, −4.212067983602419660332655034017, −3.12334482551785126887163851205, −2.25425217775524356163092392093, −1.20240905852378992152577530358, −0.74293075111814831528987161708, 1.01800170823373969825697609386, 1.46710874938487506242725071677, 2.79276928462720894795639330712, 3.32805528073450395523604929493, 4.0973399889475453000456666511, 5.39309498354407044364426968002, 5.89599309871059650670163896947, 6.60809612913612446242748912096, 7.3077844487473457403635516208, 8.410400408287709042444241941985, 8.82724153136035574126131172143, 9.94527364269529819216940127906, 10.35155344677135719365554135879, 11.19368140978765991322131436715, 11.76812176882387494069166065570, 12.71133869655733219512313125235, 13.63678991551875331974551770047, 13.933249852512113953613250168905, 14.71950093143821157484872511734, 15.43977700359148311781729549388, 16.23122593609100789405183369266, 16.96202689844030479671077244150, 17.65508079259894567682852124482, 18.23820932997872432626684818973, 19.156002296142705275271646779482

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