Properties

Label 2-2400-1.1-c1-0-23
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 − 2·7-s + 9-s − 6·11-s + 2·13-s + 6·17-s + 4·19-s + 2·21-s + 8·23-s − 27-s − 8·31-s + 6·33-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 4·47-s − 3·49-s − 6·51-s − 6·53-s − 4·57-s − 6·59-s − 6·61-s − 2·63-s − 8·69-s − 4·71-s − 12·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s − 0.192·27-s − 1.43·31-s + 1.04·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s − 0.768·61-s − 0.251·63-s − 0.963·69-s − 0.474·71-s − 1.40·73-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−8.581121196771271728980142550881, −7.58143893892209121895073378690, −7.20207685475442227431828586556, −6.08198532619333796070347285722, −5.41390091613278019584528777631, −4.89368686770547847465596752720, −3.41622971494981084233413281923, −2.95921669963291858838860415843, −1.37456716929394551516171235179, 0, 1.37456716929394551516171235179, 2.95921669963291858838860415843, 3.41622971494981084233413281923, 4.89368686770547847465596752720, 5.41390091613278019584528777631, 6.08198532619333796070347285722, 7.20207685475442227431828586556, 7.58143893892209121895073378690, 8.581121196771271728980142550881

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