Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \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  − 7-s − 4·11-s + 13-s + 4·17-s + 19-s − 4·23-s − 4·29-s − 5·31-s + 6·37-s − 12·41-s − 5·43-s + 8·47-s − 6·49-s − 12·53-s − 8·59-s + 7·61-s − 13·67-s − 12·71-s + 6·73-s + 4·77-s + 12·79-s − 8·83-s − 91-s + 13·97-s − 12·101-s + 4·103-s + 12·107-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s − 0.834·23-s − 0.742·29-s − 0.898·31-s + 0.986·37-s − 1.87·41-s − 0.762·43-s + 1.16·47-s − 6/7·49-s − 1.64·53-s − 1.04·59-s + 0.896·61-s − 1.58·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-s + 1.35·79-s − 0.878·83-s − 0.104·91-s + 1.31·97-s − 1.19·101-s + 0.394·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1800} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1800,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.895716002168851437744218824241, −7.929415382220924128214461090885, −7.52145300864874585303566247826, −6.39754463361252039054477440603, −5.63705675785723559506487595163, −4.89225670237683858046256523275, −3.69114066821144829351188843471, −2.92281462122494924475120025499, −1.68391378455870970336182945013, 0, 1.68391378455870970336182945013, 2.92281462122494924475120025499, 3.69114066821144829351188843471, 4.89225670237683858046256523275, 5.63705675785723559506487595163, 6.39754463361252039054477440603, 7.52145300864874585303566247826, 7.929415382220924128214461090885, 8.895716002168851437744218824241

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