Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{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  + 2-s + 4-s + 2·7-s + 8-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 4·19-s + 6·23-s − 4·26-s + 2·28-s + 6·29-s + 8·31-s + 32-s + 6·34-s + 10·37-s + 4·38-s + 6·41-s + 8·43-s + 6·46-s − 6·47-s − 3·49-s − 4·52-s + 2·56-s + 6·58-s − 8·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 0.784·26-s + 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.554·52-s + 0.267·56-s + 0.787·58-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(54450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{54450} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 54450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.957134636$
$L(\frac12)$  $\approx$  $5.957134636$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.37514846726047, −14.10275168100768, −13.46409878900677, −12.83366977895796, −12.38084945797546, −11.94998983113647, −11.45648809665936, −11.02709669956321, −10.31190512827768, −9.812470020589678, −9.436543109325477, −8.581637679820715, −7.971850161973466, −7.512507897144904, −7.208932209883659, −6.268553513575817, −5.902373607295450, −5.081091218821238, −4.797553525165767, −4.330281889459990, −3.336976517692873, −2.868728345793669, −2.355707318640834, −1.244748463243351, −0.8679066291062329, 0.8679066291062329, 1.244748463243351, 2.355707318640834, 2.868728345793669, 3.336976517692873, 4.330281889459990, 4.797553525165767, 5.081091218821238, 5.902373607295450, 6.268553513575817, 7.208932209883659, 7.512507897144904, 7.971850161973466, 8.581637679820715, 9.436543109325477, 9.812470020589678, 10.31190512827768, 11.02709669956321, 11.45648809665936, 11.94998983113647, 12.38084945797546, 12.83366977895796, 13.46409878900677, 14.10275168100768, 14.37514846726047

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