Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 2·7-s + 9-s + 11-s + 2·13-s − 2·15-s + 4·17-s + 2·21-s − 25-s − 27-s + 8·29-s + 8·31-s − 33-s − 4·35-s − 10·37-s − 2·39-s − 8·41-s − 2·43-s + 2·45-s − 8·47-s − 3·49-s − 4·51-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s + 0.970·17-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47652 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47652 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(47652\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{47652} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 47652,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.092436414$
$L(\frac12)$  $\approx$  $2.092436414$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.49928018634908, −13.87273452743896, −13.55950583299596, −13.18364966344240, −12.39684789501320, −11.96476926198673, −11.74801813523446, −10.74022832053055, −10.39179528186433, −9.915485765076417, −9.581879688157541, −8.842540270089539, −8.306536416806628, −7.710692200926793, −6.784823308949462, −6.488843557735878, −6.125159846443922, −5.392462868265561, −4.953158326219969, −4.221243989706868, −3.296769444483808, −3.071254208839258, −1.947305483749334, −1.391343434902428, −0.5508987710360389, 0.5508987710360389, 1.391343434902428, 1.947305483749334, 3.071254208839258, 3.296769444483808, 4.221243989706868, 4.953158326219969, 5.392462868265561, 6.125159846443922, 6.488843557735878, 6.784823308949462, 7.710692200926793, 8.306536416806628, 8.842540270089539, 9.581879688157541, 9.915485765076417, 10.39179528186433, 10.74022832053055, 11.74801813523446, 11.96476926198673, 12.39684789501320, 13.18364966344240, 13.55950583299596, 13.87273452743896, 14.49928018634908

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