Properties

Label 2-4200-1.1-c1-0-43
Degree $2$
Conductor $4200$
Sign $-1$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
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  − 3-s + 7-s + 9-s + 2·11-s − 2·13-s − 6·19-s − 21-s − 27-s − 6·29-s + 10·31-s − 2·33-s + 2·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 6·53-s + 6·57-s − 6·61-s + 63-s + 4·67-s + 6·71-s − 14·73-s + 2·77-s + 4·79-s + 81-s + 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.37·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 1.79·31-s − 0.348·33-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.794·57-s − 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.712·71-s − 1.63·73-s + 0.227·77-s + 0.450·79-s + 1/9·81-s + 0.643·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(33.5371\)
Root analytic conductor: \(5.79112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.066972018398009382545409531527, −7.23625883041751627947701933234, −6.46406809871360528459208750774, −5.97846627400020268994571865083, −4.88472199467215877321611461730, −4.47461775871016923541940664416, −3.48606836449520796464757548603, −2.32220817433425281855736791143, −1.38085273658773050334628712304, 0, 1.38085273658773050334628712304, 2.32220817433425281855736791143, 3.48606836449520796464757548603, 4.47461775871016923541940664416, 4.88472199467215877321611461730, 5.97846627400020268994571865083, 6.46406809871360528459208750774, 7.23625883041751627947701933234, 8.066972018398009382545409531527

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