Properties

Label 2-24150-1.1-c1-0-69
Degree $2$
Conductor $24150$
Sign $1$
Analytic cond. $192.838$
Root an. cond. $13.8866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 4·11-s − 12-s − 4·13-s − 14-s + 16-s − 4·17-s − 18-s − 2·19-s − 21-s + 4·22-s + 23-s + 24-s + 4·26-s − 27-s + 28-s − 6·29-s − 4·31-s − 32-s + 4·33-s + 4·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.696·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(192.838\)
Root analytic conductor: \(13.8866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 24150,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.12228631422580, −15.32897298627087, −15.07569227523998, −14.63385817940312, −13.59609423145840, −13.21987931648310, −12.68593848444495, −11.98775438850468, −11.58090726236866, −10.90965647434833, −10.56107388447220, −9.978236547946973, −9.479749700169454, −8.678674189679343, −8.265991850419862, −7.548007423967559, −7.048506621416176, −6.606781870305543, −5.633521736378285, −5.208623317703070, −4.663347879436072, −3.781915481525494, −2.836686741026099, −2.141221353419040, −1.532602037435628, 0, 0, 1.532602037435628, 2.141221353419040, 2.836686741026099, 3.781915481525494, 4.663347879436072, 5.208623317703070, 5.633521736378285, 6.606781870305543, 7.048506621416176, 7.548007423967559, 8.265991850419862, 8.678674189679343, 9.479749700169454, 9.978236547946973, 10.56107388447220, 10.90965647434833, 11.58090726236866, 11.98775438850468, 12.68593848444495, 13.21987931648310, 13.59609423145840, 14.63385817940312, 15.07569227523998, 15.32897298627087, 16.12228631422580

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