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Properties

Label 2-3024-84.83-c0-0-2
Degree $2$
Conductor $3024$
Sign $1$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·5-s + 7-s + 1.73·11-s − 19-s − 1.73·23-s + 1.99·25-s + 31-s − 1.73·35-s + 37-s + 1.73·41-s + 49-s − 2.99·55-s + 1.73·71-s + 1.73·77-s + 1.73·89-s + 1.73·95-s + 103-s − 109-s + 2.99·115-s + ⋯
L(s)  = 1  − 1.73·5-s + 7-s + 1.73·11-s − 19-s − 1.73·23-s + 1.99·25-s + 31-s − 1.73·35-s + 37-s + 1.73·41-s + 49-s − 2.99·55-s + 1.73·71-s + 1.73·77-s + 1.73·89-s + 1.73·95-s + 103-s − 109-s + 2.99·115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (3023, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080142307\)
\(L(\frac12)\) \(\approx\) \(1.080142307\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 + 1.73T + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - 1.73T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.698466501218923909398286939845, −8.064562056161533763852054846046, −7.65624574948318248223168597704, −6.69269449631046119413582295728, −6.02512571756188674850989997082, −4.65246744096360991301633470682, −4.17756867160781440536768687094, −3.68733327093475867208919673546, −2.26746686116074668433420346728, −0.980432780133230278840017852518, 0.980432780133230278840017852518, 2.26746686116074668433420346728, 3.68733327093475867208919673546, 4.17756867160781440536768687094, 4.65246744096360991301633470682, 6.02512571756188674850989997082, 6.69269449631046119413582295728, 7.65624574948318248223168597704, 8.064562056161533763852054846046, 8.698466501218923909398286939845

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