Properties

Degree $2$
Conductor $1200$
Sign $-1$
Motivic weight $1$
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 − 2·11-s − 2·13-s − 6·17-s − 8·19-s − 2·21-s − 4·23-s + 27-s + 8·29-s − 2·33-s + 10·37-s − 2·39-s + 2·41-s − 12·43-s − 3·49-s − 6·51-s − 10·53-s − 8·57-s + 6·59-s + 2·61-s − 2·63-s − 8·67-s − 4·69-s + 4·71-s − 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s + 0.192·27-s + 1.48·29-s − 0.348·33-s + 1.64·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 3/7·49-s − 0.840·51-s − 1.37·53-s − 1.05·57-s + 0.781·59-s + 0.256·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s + 0.474·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -1)\)

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

−9.343061317538849855270026145596, −8.473810258862941042294031254575, −7.900639168738143112286151233174, −6.66208513585632291920486678528, −6.31249851341993462828303401256, −4.82757705586149563119321113004, −4.12182189313281906026607987035, −2.86526191126796724123303244993, −2.11268904766511061908179462251, 0, 2.11268904766511061908179462251, 2.86526191126796724123303244993, 4.12182189313281906026607987035, 4.82757705586149563119321113004, 6.31249851341993462828303401256, 6.66208513585632291920486678528, 7.900639168738143112286151233174, 8.473810258862941042294031254575, 9.343061317538849855270026145596

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