Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 19^{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  − 2·3-s − 2·7-s + 9-s + 2·13-s + 6·17-s + 4·21-s − 6·23-s + 4·27-s − 6·29-s + 4·31-s + 2·37-s − 4·39-s − 6·41-s + 10·43-s + 6·47-s − 3·49-s − 12·51-s − 6·53-s − 12·59-s + 2·61-s − 2·63-s + 2·67-s + 12·69-s + 12·71-s − 2·73-s − 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s + 1.42·71-s − 0.234·73-s − 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36100\)    =    \(2^{2} \cdot 5^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{36100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 36100,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
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

−15.49774404899198, −14.48179522890825, −14.16856209370504, −13.61245115783339, −12.83514181328348, −12.51921566817818, −12.02090388977209, −11.50329876311137, −11.04107095235103, −10.33837861704804, −10.06870523182061, −9.398866521945697, −8.862669328017785, −7.946805593009705, −7.701563645291269, −6.806063579786096, −6.272196260757473, −5.849803824146301, −5.453873457762236, −4.690208479323645, −3.937677558970108, −3.375263162050059, −2.650834581082384, −1.622442630699115, −0.8215993205563163, 0, 0.8215993205563163, 1.622442630699115, 2.650834581082384, 3.375263162050059, 3.937677558970108, 4.690208479323645, 5.453873457762236, 5.849803824146301, 6.272196260757473, 6.806063579786096, 7.701563645291269, 7.946805593009705, 8.862669328017785, 9.398866521945697, 10.06870523182061, 10.33837861704804, 11.04107095235103, 11.50329876311137, 12.02090388977209, 12.51921566817818, 12.83514181328348, 13.61245115783339, 14.16856209370504, 14.48179522890825, 15.49774404899198

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