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  − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s − 6·29-s + 8·31-s − 6·37-s + 6·41-s − 4·43-s − 7·49-s − 2·53-s − 4·59-s − 2·61-s + 4·67-s − 8·71-s − 10·73-s − 8·79-s − 4·83-s + 6·89-s − 2·97-s + 18·101-s − 16·103-s − 12·107-s − 2·109-s + 18·113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.203·97-s + 1.79·101-s − 1.57·103-s − 1.16·107-s − 0.191·109-s + 1.69·113-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 \( 1 + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.730633394111998328725718228031, −8.095549656698690551636863836910, −7.47308398754126662115910632702, −6.33548194788761977078999997603, −5.74245729062321480161908713186, −4.76184067616314722575220108003, −3.84771560491769769437106298900, −2.79864903411763483594224581992, −1.72979115909686153918413689995, 0, 1.72979115909686153918413689995, 2.79864903411763483594224581992, 3.84771560491769769437106298900, 4.76184067616314722575220108003, 5.74245729062321480161908713186, 6.33548194788761977078999997603, 7.47308398754126662115910632702, 8.095549656698690551636863836910, 8.730633394111998328725718228031

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