Properties

Label 1-3024-3024.131-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.661 - 0.749i$
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  + (0.984 − 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.642 − 0.766i)13-s + 17-s i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.642 − 0.766i)29-s + (−0.939 − 0.342i)31-s + (0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)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 + (−0.642 − 0.766i)13-s + 17-s i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.642 − 0.766i)29-s + (−0.939 − 0.342i)31-s + (0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5222192772 - 1.157366458i\)
\(L(\frac12)\) \(\approx\) \(0.5222192772 - 1.157366458i\)
\(L(1)\) \(\approx\) \(1.082041594 - 0.1322580171i\)
\(L(1)\) \(\approx\) \(1.082041594 - 0.1322580171i\)

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 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (0.642 - 0.766i)T \)
89 \( 1 - T \)
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

−18.90861111379202662160821086171, −18.44273378618886864090420289069, −17.75059821610406828748261022514, −17.11648896474140669152493264599, −16.350986914438122898647449006215, −15.76863052767456927684507244187, −14.66644210383993788142883906044, −14.308994952999579085125347223242, −13.57266584358565515399371385295, −12.74466883666470981286142779297, −12.28965418804729796153464964559, −11.18967658107304397165822729153, −10.55383315755598017871811873698, −9.834407858088244832673010418332, −9.30820494854029137463783808129, −8.41417551372021567026639351450, −7.516261593074705627945714062041, −6.83256014722022826598210505711, −6.06310998820305043897178352276, −5.17609725575665828108123637220, −4.74150097153861719904548441702, −3.50568046468511121942691949590, −2.56843090790799635949224881529, −2.06833080618904102356527827291, −0.97665104489029178133710830020, 0.20193092566211427212881738152, 1.23920238765130768020092104944, 2.165518336249550912095197320199, 2.8756368325607269152061176507, 3.78012289766591003350597109107, 4.91717737856488388122574323923, 5.61427801568508539671410701365, 5.942569124467506892044792359201, 7.12704301670156441779574820801, 7.9216850804992929746296644290, 8.42114398563514262879717583608, 9.63768212367589960531731269440, 10.02492661559717235885695027779, 10.51830196961161025410261783531, 11.63741742769438808722693871510, 12.38106320388911272325067912505, 13.03169341926421954331172524469, 13.659122380099860061389469352783, 14.376194496394809129866812198478, 15.05864450314299539386547655300, 15.88052285635112154840958209978, 16.68108454428755768576259420612, 17.15066963531324587441420928508, 18.04382583391272274128266196712, 18.435921403559288650082929916720

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