Properties

Degree 2
Conductor $ 2^{2} \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  − 2·7-s − 2·13-s − 6·17-s − 4·19-s + 6·23-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s − 6·53-s − 12·59-s + 2·61-s − 2·67-s + 12·71-s − 2·73-s + 8·79-s + 6·83-s + 6·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s + 0.658·83-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 900,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−19.31331797455429, −18.85646433680024, −17.91302395987430, −17.18051775404703, −16.64685884649347, −15.74261731841818, −15.14645439460621, −14.46161299971597, −13.37025234542131, −12.94966976872364, −12.22851421321623, −11.08138326769701, −10.70077912534757, −9.447689111417191, −9.113263281253577, −8.008294857926527, −6.961670916910435, −6.418423898307642, −5.248544354531793, −4.285303654324059, −3.168079674272959, −2.018362332160805, 0, 2.018362332160805, 3.168079674272959, 4.285303654324059, 5.248544354531793, 6.418423898307642, 6.961670916910435, 8.008294857926527, 9.113263281253577, 9.447689111417191, 10.70077912534757, 11.08138326769701, 12.22851421321623, 12.94966976872364, 13.37025234542131, 14.46161299971597, 15.14645439460621, 15.74261731841818, 16.64685884649347, 17.18051775404703, 17.91302395987430, 18.85646433680024, 19.31331797455429

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