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  + 3·7-s − 2·11-s − 3·13-s − 6·17-s − 7·19-s − 6·23-s + 2·29-s − 5·31-s + 10·37-s − 12·41-s + 3·43-s + 10·47-s + 2·49-s + 6·59-s − 13·61-s + 7·67-s + 4·71-s − 6·73-s − 6·77-s − 8·79-s + 6·83-s − 16·89-s − 9·91-s − 7·97-s − 12·103-s + 16·107-s + 9·109-s + ⋯
L(s)  = 1  + 1.13·7-s − 0.603·11-s − 0.832·13-s − 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.371·29-s − 0.898·31-s + 1.64·37-s − 1.87·41-s + 0.457·43-s + 1.45·47-s + 2/7·49-s + 0.781·59-s − 1.66·61-s + 0.855·67-s + 0.474·71-s − 0.702·73-s − 0.683·77-s − 0.900·79-s + 0.658·83-s − 1.69·89-s − 0.943·91-s − 0.710·97-s − 1.18·103-s + 1.54·107-s + 0.862·109-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.687317716187599920863113173278, −8.185950120514054282770297981763, −7.39251670550150742087066701310, −6.51300867975066392569978306528, −5.58239379032457876412961380567, −4.62488497702715547946506468996, −4.15167539162411011404326170827, −2.52407450006420425610946316785, −1.88730569082121758267088569211, 0, 1.88730569082121758267088569211, 2.52407450006420425610946316785, 4.15167539162411011404326170827, 4.62488497702715547946506468996, 5.58239379032457876412961380567, 6.51300867975066392569978306528, 7.39251670550150742087066701310, 8.185950120514054282770297981763, 8.687317716187599920863113173278

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