Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 27-s − 2·29-s − 4·33-s + 10·37-s − 2·39-s + 10·41-s + 4·43-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s + 4·57-s + 4·59-s − 2·61-s + 12·67-s + 8·71-s − 10·73-s + 81-s + 12·83-s + 2·87-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 1/9·81-s + 1.31·83-s + 0.214·87-s − 0.635·89-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  =  0
Selberg data  =  $(2,\ 1200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.427722446$
$L(\frac12)$  $\approx$  $1.427722446$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.655893575783761174602693945437, −9.049118048765622993735733025092, −8.151425844540532890281788534319, −7.10478788988449333222322211892, −6.34199635143511529870167101224, −5.72078508715635747921690343280, −4.42801570520906772580093162574, −3.86223113294681320091712334301, −2.32158421366848578869138486097, −0.957897388351054006923114027189, 0.957897388351054006923114027189, 2.32158421366848578869138486097, 3.86223113294681320091712334301, 4.42801570520906772580093162574, 5.72078508715635747921690343280, 6.34199635143511529870167101224, 7.10478788988449333222322211892, 8.151425844540532890281788534319, 9.049118048765622993735733025092, 9.655893575783761174602693945437

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