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Properties

Label 2-3024-7.5-c0-0-1
Degree $2$
Conductor $3024$
Sign $-0.126 + 0.991i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (−0.5 − 0.866i)7-s + (−1.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (1 − 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (1 − 1.73i)79-s − 1.73i·97-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−1.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (1 − 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (1 − 1.73i)79-s − 1.73i·97-s + (−0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.126 + 0.991i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ -0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8885928569\)
\(L(\frac12)\) \(\approx\) \(0.8885928569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.648945698951114820520293425504, −7.965417739014474246010891126742, −7.15165673939126676052198142380, −6.44975179771677594213385084891, −5.85022465390939629681901195272, −4.49439022163802085464188620348, −4.20535541113172442382398115999, −3.02922798616085085357130631764, −2.09677632069388811530758308037, −0.53659147058404712339798999000, 1.57661444517640536435648763308, 2.63365814456375771932375669693, 3.45997084599980878527322066409, 4.48827454076173295228232081585, 5.29546437456800243625744192685, 6.33048902036013736782946845390, 6.49293798176864093981901029232, 7.80051716747258995739023934599, 8.362177901967644800640795167687, 9.054177946641286646910082250703

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