Properties

Label 2-3024-1.1-c1-0-13
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  − 5-s + 7-s + 5·11-s − 2·17-s + 19-s − 23-s − 4·25-s − 4·29-s + 9·31-s − 35-s + 5·37-s + 9·41-s + 10·43-s + 6·47-s + 49-s − 12·53-s − 5·55-s − 14·59-s + 8·67-s − 13·71-s − 2·73-s + 5·77-s − 6·79-s − 4·83-s + 2·85-s + 9·89-s − 95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.50·11-s − 0.485·17-s + 0.229·19-s − 0.208·23-s − 4/5·25-s − 0.742·29-s + 1.61·31-s − 0.169·35-s + 0.821·37-s + 1.40·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s − 1.64·53-s − 0.674·55-s − 1.82·59-s + 0.977·67-s − 1.54·71-s − 0.234·73-s + 0.569·77-s − 0.675·79-s − 0.439·83-s + 0.216·85-s + 0.953·89-s − 0.102·95-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: $\chi_{3024} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.915352400\)
\(L(\frac12)\) \(\approx\) \(1.915352400\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.852235364105462314758322151711, −7.82180523610847379974298804156, −7.41731738918212721913056577334, −6.31189353147311880492962393494, −5.91137064624602427944931283698, −4.50854174640145518647490777803, −4.21595600702686973203131728090, −3.16207871637142803917205041928, −1.99414709782610576510843414106, −0.875062595129075836979928350629, 0.875062595129075836979928350629, 1.99414709782610576510843414106, 3.16207871637142803917205041928, 4.21595600702686973203131728090, 4.50854174640145518647490777803, 5.91137064624602427944931283698, 6.31189353147311880492962393494, 7.41731738918212721913056577334, 7.82180523610847379974298804156, 8.852235364105462314758322151711

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