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Properties

Label 2-3024-1.1-c1-0-33
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 6·13-s + 7·17-s + 8·19-s − 2·23-s − 4·25-s − 4·29-s + 2·31-s + 35-s + 9·37-s − 11·41-s + 7·43-s − 11·47-s + 49-s − 10·53-s + 9·59-s − 8·61-s + 6·65-s − 12·71-s − 14·73-s − 79-s − 9·83-s − 7·85-s − 14·89-s + 6·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.66·13-s + 1.69·17-s + 1.83·19-s − 0.417·23-s − 4/5·25-s − 0.742·29-s + 0.359·31-s + 0.169·35-s + 1.47·37-s − 1.71·41-s + 1.06·43-s − 1.60·47-s + 1/7·49-s − 1.37·53-s + 1.17·59-s − 1.02·61-s + 0.744·65-s − 1.42·71-s − 1.63·73-s − 0.112·79-s − 0.987·83-s − 0.759·85-s − 1.48·89-s + 0.628·91-s − 0.820·95-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.057394536689059727049816407146, −7.62814961683791324343261012652, −7.09719034829237047645339493045, −5.92355118351395198688122554211, −5.31750687150200547436919808681, −4.46145248722790846311222725619, −3.40313836526304898756023679034, −2.81096596415057142508365378348, −1.42053801208435309098292606008, 0, 1.42053801208435309098292606008, 2.81096596415057142508365378348, 3.40313836526304898756023679034, 4.46145248722790846311222725619, 5.31750687150200547436919808681, 5.92355118351395198688122554211, 7.09719034829237047645339493045, 7.62814961683791324343261012652, 8.057394536689059727049816407146

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