Properties

Label 2-3024-1.1-c1-0-26
Degree $2$
Conductor $3024$
Sign $1$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 7-s + 4·11-s + 3·13-s − 7·17-s − 2·19-s + 23-s + 11·25-s + 29-s + 9·31-s + 4·35-s + 2·37-s + 6·41-s − 11·43-s + 6·47-s + 49-s − 9·53-s + 16·55-s + 5·59-s − 6·61-s + 12·65-s − 7·67-s + 7·71-s − 14·73-s + 4·77-s + 6·79-s + 4·83-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.377·7-s + 1.20·11-s + 0.832·13-s − 1.69·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s + 0.185·29-s + 1.61·31-s + 0.676·35-s + 0.328·37-s + 0.937·41-s − 1.67·43-s + 0.875·47-s + 1/7·49-s − 1.23·53-s + 2.15·55-s + 0.650·59-s − 0.768·61-s + 1.48·65-s − 0.855·67-s + 0.830·71-s − 1.63·73-s + 0.455·77-s + 0.675·79-s + 0.439·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3024} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.068115494\)
\(L(\frac12)\) \(\approx\) \(3.068115494\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.906061337314963241993527708313, −8.227259731881334231162852635498, −6.83264370513871148909510884497, −6.42027135383899144087751394583, −5.89795561504121848553653635959, −4.83292001561835341457693151881, −4.18186575003893897709757763846, −2.85236525643673920019681416598, −1.95757589543939353982506678305, −1.19950554795905379248633413151, 1.19950554795905379248633413151, 1.95757589543939353982506678305, 2.85236525643673920019681416598, 4.18186575003893897709757763846, 4.83292001561835341457693151881, 5.89795561504121848553653635959, 6.42027135383899144087751394583, 6.83264370513871148909510884497, 8.227259731881334231162852635498, 8.906061337314963241993527708313

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