Properties

Label 2-2400-1.1-c1-0-31
Degree $2$
Conductor $2400$
Sign $-1$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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 − 4·7-s + 9-s + 4·13-s − 8·19-s − 4·21-s + 4·23-s + 27-s − 6·29-s − 8·31-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 4·47-s + 9·49-s − 12·53-s − 8·57-s − 6·61-s − 4·63-s + 12·67-s + 4·69-s − 16·71-s − 8·79-s + 81-s − 12·83-s − 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s − 1.64·53-s − 1.05·57-s − 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s − 1.89·71-s − 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.902823429730034903796185580495, −7.84161812555699504527813747179, −6.95756593027595618574514129256, −6.33937261327383596982485750252, −5.63085870340990006632960124727, −4.27460411019146365757900718473, −3.60405203154249968708316901683, −2.84502227093973034349885876665, −1.68377693849669875173825025336, 0, 1.68377693849669875173825025336, 2.84502227093973034349885876665, 3.60405203154249968708316901683, 4.27460411019146365757900718473, 5.63085870340990006632960124727, 6.33937261327383596982485750252, 6.95756593027595618574514129256, 7.84161812555699504527813747179, 8.902823429730034903796185580495

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