Properties

Label 2-3024-1.1-c1-0-12
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  − 7-s + 5·13-s + 3·17-s − 2·19-s + 9·23-s − 5·25-s − 3·29-s − 5·31-s + 2·37-s − 6·41-s + 43-s + 6·47-s + 49-s + 3·53-s + 3·59-s − 10·61-s + 13·67-s − 9·71-s + 2·73-s + 10·79-s + 12·83-s + 15·89-s − 5·91-s + 8·97-s + 18·101-s + 13·103-s + 12·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.38·13-s + 0.727·17-s − 0.458·19-s + 1.87·23-s − 25-s − 0.557·29-s − 0.898·31-s + 0.328·37-s − 0.937·41-s + 0.152·43-s + 0.875·47-s + 1/7·49-s + 0.412·53-s + 0.390·59-s − 1.28·61-s + 1.58·67-s − 1.06·71-s + 0.234·73-s + 1.12·79-s + 1.31·83-s + 1.58·89-s − 0.524·91-s + 0.812·97-s + 1.79·101-s + 1.28·103-s + 1.16·107-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

\(L(1)\) \(\approx\) \(1.873892735\)
\(L(\frac12)\) \(\approx\) \(1.873892735\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−8.891978601849373046435074140087, −7.949595272551558511484925825699, −7.24729027002187867605972756035, −6.38983991712328031863989081821, −5.75498451669473966849176010933, −4.92409316048517899781293445732, −3.77842535731702409100199368470, −3.29476397189543990973285365054, −2.01879358291610012679245055865, −0.856193565353031792325176418947, 0.856193565353031792325176418947, 2.01879358291610012679245055865, 3.29476397189543990973285365054, 3.77842535731702409100199368470, 4.92409316048517899781293445732, 5.75498451669473966849176010933, 6.38983991712328031863989081821, 7.24729027002187867605972756035, 7.949595272551558511484925825699, 8.891978601849373046435074140087

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