Properties

Label 2-3024-1.1-c1-0-11
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

\(L(1)\) \(\approx\) \(1.944950498\)
\(L(\frac12)\) \(\approx\) \(1.944950498\)
\(L(\frac{3}{2})\) 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

−8.478426573102990555594665044264, −8.098584445070679785979739673973, −7.30779100190829260650015981954, −6.41579425095070423565731763913, −5.51926476472048359134068258767, −5.05061740660147245068087666602, −4.05627857283460152298243124357, −2.88260927317063896946056802272, −2.20921145148586991820085196748, −0.853219031875925699013328984683, 0.853219031875925699013328984683, 2.20921145148586991820085196748, 2.88260927317063896946056802272, 4.05627857283460152298243124357, 5.05061740660147245068087666602, 5.51926476472048359134068258767, 6.41579425095070423565731763913, 7.30779100190829260650015981954, 8.098584445070679785979739673973, 8.478426573102990555594665044264

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