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Properties

Label 2-3024-84.83-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(1.678892697\)
\(L(\frac12)\) \(\approx\) \(1.678892697\)
\(L(1)\) 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

−9.220934397059300607540249598186, −8.347728992262959699360247183095, −7.09788539494229225262094300827, −6.51203353419342999188611143878, −5.96981051688074605029205555994, −5.33708095366491303049895833301, −4.08841682940364961629247717656, −3.29199085407115824441051380853, −2.18507256971019530681992864192, −1.33404683320225206358070785303, 1.33404683320225206358070785303, 2.18507256971019530681992864192, 3.29199085407115824441051380853, 4.08841682940364961629247717656, 5.33708095366491303049895833301, 5.96981051688074605029205555994, 6.51203353419342999188611143878, 7.09788539494229225262094300827, 8.347728992262959699360247183095, 9.220934397059300607540249598186

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