Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} $
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 − 4·7-s + 9-s − 2·13-s − 6·17-s + 4·19-s − 4·21-s + 27-s − 6·29-s − 8·31-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 9·49-s − 6·51-s + 6·53-s + 4·57-s − 10·61-s − 4·63-s − 4·67-s − 2·73-s − 8·79-s + 81-s + 12·83-s − 6·87-s + 18·89-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 1.90·89-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

\( d \)  =  \(2\)
\( N \)  =  \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1200} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.276841907117526756164869085518, −8.811879039450229959581280002555, −7.53023481714424969283045906095, −6.97121259233254998318201926571, −6.10897773314734163440451778623, −5.03186235211519720784151262088, −3.82839635062702052518767893213, −3.10714236804866821433728477742, −2.01462355044529552405323097963, 0, 2.01462355044529552405323097963, 3.10714236804866821433728477742, 3.82839635062702052518767893213, 5.03186235211519720784151262088, 6.10897773314734163440451778623, 6.97121259233254998318201926571, 7.53023481714424969283045906095, 8.811879039450229959581280002555, 9.276841907117526756164869085518

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