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

−8.772421839183462887904267277344, −8.279043826836005007614143835778, −7.14524835419643162911975279792, −6.53690929538739447537109821251, −5.70088645188412764957726143386, −4.85182738731759099554593417847, −3.55277063573082460622069589535, −3.08865327644648973893039079227, −1.63004456496430691453589862078, 0, 1.63004456496430691453589862078, 3.08865327644648973893039079227, 3.55277063573082460622069589535, 4.85182738731759099554593417847, 5.70088645188412764957726143386, 6.53690929538739447537109821251, 7.14524835419643162911975279792, 8.279043826836005007614143835778, 8.772421839183462887904267277344

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