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Properties

Label 2-3024-9.4-c1-0-13
Degree $2$
Conductor $3024$
Sign $0.686 - 0.726i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.300 − 0.520i)5-s + (−0.5 − 0.866i)7-s + (0.800 + 1.38i)11-s + (−0.165 + 0.286i)13-s + 1.44·17-s − 2.57·19-s + (−0.924 + 1.60i)23-s + (2.31 + 4.01i)25-s + (1.75 + 3.04i)29-s + (−4.81 + 8.33i)31-s − 0.600·35-s + 0.600·37-s + (3.31 − 5.73i)41-s + (1.81 + 3.14i)43-s + (−1.95 − 3.38i)47-s + ⋯
L(s)  = 1  + (0.134 − 0.232i)5-s + (−0.188 − 0.327i)7-s + (0.241 + 0.417i)11-s + (−0.0458 + 0.0793i)13-s + 0.351·17-s − 0.591·19-s + (−0.192 + 0.333i)23-s + (0.463 + 0.803i)25-s + (0.326 + 0.565i)29-s + (−0.863 + 1.49i)31-s − 0.101·35-s + 0.0987·37-s + (0.516 − 0.895i)41-s + (0.276 + 0.479i)43-s + (−0.285 − 0.493i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.686 - 0.726i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.686 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.632655791\)
\(L(\frac12)\) \(\approx\) \(1.632655791\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.300 + 0.520i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.800 - 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.165 - 0.286i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
23 \( 1 + (0.924 - 1.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.81 - 8.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.600T + 37T^{2} \)
41 \( 1 + (-3.31 + 5.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.81 - 3.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.27T + 53T^{2} \)
59 \( 1 + (-6.93 + 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.90 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 - 4.13T + 73T^{2} \)
79 \( 1 + (-4.06 - 7.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.78 - 4.83i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.83T + 89T^{2} \)
97 \( 1 + (-0.974 - 1.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.892596662139224429130597811022, −8.120894730248112530526752816999, −7.13448589745513693827097032716, −6.78491012403317602356731485696, −5.65692671646363903426138422452, −5.03807756662903995009808695981, −4.05976273649236523288124540529, −3.31341875190398399814795481868, −2.12201334900233135896083199526, −1.07267093840228267993026118255, 0.58560841705053492336758568031, 2.07112994220007601852262094043, 2.87439146887436528678278164299, 3.90013848681340294555886475379, 4.67481309907706848567715027057, 5.83902807465235604717133299478, 6.16752046918286805527832265742, 7.11658074155615669398863657002, 7.931367962072400828018550817862, 8.628900704132766548508420104526

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